Chapter 1-2 Add and Subtract Integers

Transcription

1 Chapter 1-2 Add and Subtract Integers Absolute Value of a number is its distance from zero on the number line. 5 = 5 and 5 = 5 Adding Numbers with the Same Sign: Add the absolute values and use the sign of the original numbers = 9 and 2 + ( 9) = 11 Adding Numbers with Different Signs: Subtract the absolute values and use the sign of the original number with the greater absolute value = 4 and 3 + ( 15) = 12 Subtracting Numbers with Any Sign: To subtract a number, add its opposite and follow the rules for adding numbers = 7 + ( 10) = 3 3 ( 12) = = 9 Examples: Add ( 96) ( 54)

2 Chapter 1-3 Multiply and Divide Integers Multiplying or Dividing with the Same Sign: If two numbers have the same sign then their product or quotient is positive. 3 6 = 18 and 40 ( 5) = 8 Multiplying or Dividing Numbers with Different Signs: If two numbers have different signs then their product or quotient is negative = 96 and 30 ( 5) = 6 Property of Zero: The product of zero and any number is zero. 0 a = 0 The quotient of zero and any nonzero number is zero. 0 a = 0 Division by zero is undefined. a 0 is undefined. Examples: 1. 5 ( 4) = = 2. 9 ( 5) = ( 5) = = ( 12) =

3 Chapter 7-7 Combining Like Terms The terms of an algebraic expression are the parts that are added together. 3x 2, 5x, 3 Like terms contain the same variables with the same powers. 9x, 6x, 2x A coefficient is a number multiplied by a variable. Combining like Terms: When simplifying an algebraic expression, add or subtract the coefficients of the like terms. Example: Combine the like terms 2x x Example: Combine the like terms 2 3 x x Example: Combine the like terms. 3x 2 5x x 2 + 3x + 6

4 A monomial is a number, a variable, or a product of numbers and variables with whole number powers. 5 x 3x x3 A polynomial is a sum or difference of two or more monomials. 2x 5 x 2 + 3x 4 2x 3 x 2 + 3x 1 Adding or Subtracting Polynomials: To add or subtract polynomials, combine like terms. Example: (x 2 + 2x + 3) + (2x 2 + 5x + 6) x 2 + 2x 2 + 2x + 5x x 2 + 7x + 9 Example: (5x 3 + 2x + 9) (3x 3 + 6x + 1) (5x 3 + 2x + 9) + ( 3x 3 6x 1) 5x 3 3x 3 + 2x 6x x 3 4x + 8

5 Chapter 1-7 Distributive Property Distributive Property: a (b + c) = a b + a c or (b + c) a = a b + a c Use the distributive property to simplify an expression. 5(2x + 3) = 5 2x = 10x + 15 Complete the distributive property 3(5x + 2) = x + Complete the distributive property 6x + 8 = ( x + ) Always apply the distributive property before combining like terms when simplifying an algebraic expression.

6 Example: Simplify. 2(x 6) + 3x Example: Simplify. 3(5x + 4) 2(6x + 1) Examples: Simplify. 1. 5x + 3(2x + 4) 2. 2(3x 2) (2x + 4) + 3(3x 5) 4. 8(x + 1) 5(x 2)

7 Chapter 1-7 Evaluating Expressions Order of Operations: When simplifying a numerical expression perform the operations in the following order, 1. Perform operations inside grouping symbols. 2. Evaluate powers. 3. Perform multiplication and division from left to right. 4. Perform addition and subtraction from left to right. Example: Simplify 2(5 3) + 7 Example: Simplify 4 + 3(2 5) Example: Simplify ( ) (9 7) 2 To evaluate an expression means to replace the variables in the expression with numbers and simplify the result.

8 Evaluate, ab 2c, use a = 3, b = 4, and c = 5 (3)(4) 2(5) Evaluate, 5 + x 2 4y, use x = 5 and y = (5) 2 4(10) Example: Evaluate each expression use a = 2, b = 3, c = 5 1. a(b c) 2. c 2 + ab 3. ab 3 + c 4. abc + (a + b) 2

9 Chapter 2-1 Solving One Step Equations An equation is a mathematical statement that two expressions are equal. 5x + 2 = 17 A solution is a value of the variable that makes the equation true. Addition/Subtraction Property: The same number can be added or subtracted to both sides of an equation and the statement will still be true. If a = b, then a + c = b + c If a = b, then a c = b c Example: Solve, x 4.6 = 2.9 Example: Solve, 45 + x = 87

10 Multiplication/Division Property: The same number can be multiplied or divided on both sides of an equation and the statement will still be true. If a = b, then a c = b c If a = b, then a c = b c Example: Solve, Example: Solve, Example: Solve, Example: Solve, x 5 = 6 2x = x = x = 3 8 Examples: Solve for x, x = x = x 2 3 = 1 6

11 Chapter 2-2 Solving Two Step Equations An equation that contains two operations, like multiplication and addition, is called a two step equation. 65 = 5x + 10 How to Solve a Two Step Equation: 1. Isolate the variable term by adding or subtracting the constant number to each side of the equation. 2. Divide each side of the equation by the coefficient of the variable term. 3. Check your solution by substituting the value back into the original equation. Example: Solve and check your solution, 65 = 5x + 10 Example: Solve and check your solution, 2x 1.2 = 5.8

12 Example: Solve and check your solution, 4 3 x + 11 = 23 Example: Solve and check your solution, 7 8 = x Examples: Solve and check your solution, 1. 2x 12 = = x = 5x x = x = = 3 8 x 3 2

13 Chapter 2-3 Solving Multi-Step Equations How to Solve a Multi-Step Equation: 1. Apply the distributive property to remove any grouping symbols from the equation. 2. Combine any like terms to simplify the expressions in the equation. 3. Isolate the variable term by adding or subtracting the constant number to each side of the equation. 4. Divide each side of the equation by the coefficient of the variable term. 5. Check your solution by substituting the value back into the original equation. Example: Solve and check your solution, 2(3x 5) = 20 Example: Solve and check your solution, 12 = 8x 3(6 + x)

14 Example: Solve and check your solution, 4(x + 1) + 2(x 7) = 50 Example: Solve and check your solution, 2 (3x 6) = 8 3 Examples: Solve and check your solution, 1. 3(3x + 4) = = x 6 2x (5 + 2x) = = 3(2x 1) 3x (4x + 8) = 6

15 Chapter 2-4 Variables on Both Sides of Equation Some equations have variables on both sides of the equal sign. To solve equations like these, 1. Apply the distributive property and combine like terms, if necessary. 2. Collect the terms with variables on one side by adding or subtracting a variable term on each side of the equation. 3. Solve and check the resulting equation. Example: Solve and check your solution, 8x + 6 = 4x 14 Example: Solve and check your solution, 2(3x 4) = x Example: Solve and check your solution, 6(2x 3) 8x = 6 + 2x Example: Solve and check your solution, 3 5n + 2n = 5 2(1 n)

16 An identity is an equation that is true, no matter what value is substituted for the variable. The solution set of an identity is all real numbers. Example: Solve, 2(x + 4) 5 = 2x + 3 Some equations are always false no matter what value is substituted for the variable. These equations have no solution. Example: Solve, 3(2x 1) + 5 = 6(x + 1) Solve the following equations: 1. 6x + 8 = 4x (1 2x) = 2x x = 2(x + 2) 4. 2(2x 3) + 10 = 6(x 1)

17 Chapter 2-5 Proportions A proportion is an equation that states two ratios are equal. Cross Products Property: If a b = c d, then ad = bc Example: Solve, x 9 = 4 6 Example: Solve, 3 x 3 = 2 x Example: Solve, 2 x + 4 = 4 x 1

18 Examples: Solve each proportion = 5 x = 6 x x = 6 9 x 5. x 6 4 = x = 7 10 x = x+4 5 x+3

19 Chapter 2-6 Literal Equations An equation that contains two or more variables is called a literal equation. A = 2πr E = mc 2 d = rt How to Solve a Literal Equation for a variable: 1. Find the variable you are asked to solve for in the literal equation. 2. Identify the operations that apply to this variable. 3. Use the add/subtract or multiply/divide properties to isolate this variable. Example: Solve for the variable n, 6 8m = 2n Example: Solve for the variable b, 4a + 2b = 6 Example: Solve for the variable x, y = mx + b

20 Example: Solve for the variable h, A = bh 2 Examples: Solve for the variable y, 1. 2y + 12 = 4x 3. 5y = 10 15x 2. 3y 6x = y 2 = 2x

21 Chapter 2-7 Absolute Value Equations Absolute Value Equation: If x = a, then x = a or x = a To solve an absolute value equation, isolate the absolute value expression on one side of the equation, then apply the absolute value equation definition. Example: Solve, 3x = 15 Example: Solve, 2x 4 = 6 Example: Solve, 5x = 35 Example: Solve, 3 2x 6 8 = 10

22 Special Cases of Absolute Value Equations: If the absolute value expression equals zero, then the absolute value equation has exactly one solution. If the absolute value expression equals a negative number, then the absolute value equation has no solutions. Example: Solve, 2x = 6 Example: Solve, 3x + 4 = 2 Examples: Solve. 1. 2x + 6 = x 4 = x 6 = x = 4

23 Chapter 3-1, 3-2 Solving Inequalities by Adding An inequality is a statement that two quantities are not equal. A < B A > B "A less than B" "A greater than B" A B "A less than or equal to B" A B "A greater than or equal to B" The solutions of an inequality are the values that make the inequality true. The graph of an inequality is the set of points on the number line that are solutions of the inequality. x < a x > a a x a a x a a a Equivalent Inequalities: x < a is equivalent to a > x. x > a is equivalent to a < x.

24 Addition/Subtraction Property: The same number can be added/subtracted to both sides of an inequality and the statement will still be true. If a < b, then a + c < b + c If a < b, then a c < b c Example: Solve and graph, x 14 < 12 Example: Solve and graph, x Examples: Solve and graph, 1. x < > x x 2 3

25 Chapter 3-3 Solving Inequalities by Multiplying Positive Multiplication/Division Property: The same positive number c can be multiplied or divided to both sides of an inequality and the statement will be true. If a < b, then c a < c b If a < b, then a c < b c Example: Solve and graph, 2x 12 Example: Solve and graph, 3 > x 4

26 Negative Multiplication/Division Property: The same negative number c can be multiplied or divided to both sides of an inequality and the statement will be true if the inequality symbol is reversed. If a < b, then c a > c b If a < b, then a ( c) > b ( c) Example: Solve and graph, 4x 44 Example: Solve and graph, x 3 2

27 Two-Step Inequalities: Apply the addition/subtraction property first then apply the multiplication/division property to solve a two-step inequality. Example: Solve and graph, 5x 4 16 Example: Solve and graph, 9 > x Examples: Solve and graph, 1. 2x < > 3x x 1 9

28 Chapter 3-4 Solving Multi-Step Inequalities A multi-step inequality is an inequality that involves two or more operations (x + 4) > 3 The solutions of a multi-step inequality are the values that make the inequality true. How to Solve a Multi-Step Inequality: 1. Apply the distributive property to remove any grouping symbols from the inequality. 2. Combine any like terms to simplify the expressions in the inequality. 3. Apply the addition/subtraction property and the multiplication/division property to solve. 4. Graph the solutions on a number line Example: Solve and graph, 2(x + 4) + 3 > 3

29 Example: Solve and graph, 2 > 7x 2(x 4) Examples: Solve and graph the solutions to each inequality. 1. 4(x + 3) > (x 12) 3. 2(x 3) 4 > (2x 4)

30 Chapter 3-5 Variables on Both Sides Some inequalities have variable terms on both sides of the inequality symbol. 2(x + 4) + 3x 2x + 3 How to Solve an Inequality with Variables on Both Sides: 1. Apply the distributive property and combine like terms as in a multi-step inequality, if necessary. 2. Use the properties of inequalities to collect all the variable terms on the left side of the inequality and all constant terms on the right side of the inequality. 3. Solve the inequality and graph the solutions. Example: Solve and graph, 2(x + 4) + 3x 2x + 3 Example: Solve and graph, 5(x + 3) 6 > x + 3

31 Some inequalities are true no matter what value is substituted for the variable. For these inequalities, the solution set is all real numbers. Example: Solve, 3x (10 + x) Some inequalities are false no matter what value is substituted for the variable. These inequalities have no solution. Their solution is the empty set,. Example: Solve 4(1 x) < 2(2x + 3) Examples: Solve and graph the solutions to each inequality. 1. 5(x + 4) > 6 + 3x 2. 4(3 x) 5(x + 1) 3. 2x (5 + x) 4. 4(x + 1) > 4x + 2

32 Chapter 3-6 Solving Compound Inequalities A compound inequality is formed when inequalities are combined into one statement by the words and or or. x > a and x < b a < x < b x a and x b a x b a b a b x < a or x > b x a or x b a b a b How to Solve a Compound Inequality: 1. Solve each simple inequality separately. 2. Graph the solution to each simple inequality. 3. Take the intersection of the graphs for compound inequalities involving and. 4. Take the union of the graphs for compound inequalities involving or.

33 Example: Solve and graph, 10 2x Example: Solve and graph, 5x 10 < 25 or 3x + 9 > 27 Examples: Solve and graph each compound inequality x x 3 < 2 or x + 10 > < 5x + 10 < x or 3x 9 12

34 Chapter 3-7 Solving Absolute-Value Inequalities An absolute-value inequality is an inequality that contains an absolute-value expression. Absolute-Value Inequalities Involving Less Than: x < a means a < x < a x a means a x a -a a -a a Absolute-Value Inequalities Involving Greater Than: x > a means x < a or x > a x a means x a or x a -a a -a a

35 How to Solve an Absolute-Value Inequality: 1. Convert the absolute-value inequality to a compound inequality. 2. Solve each simple inequality separately. 3. Graph the solution to each simple inequality. 4. Take the intersection of the graphs for compound inequalities involving and. 5. Take the union of the graphs for compound inequalities involving or. Example: Solve and graph, 2x 4 8 Example: Solve and graph, 5x + 15 > 10

36 Chapter 4-2 Relations and Functions A relation is a set of ordered pairs. {(2, 3), ( 1, 5), (4, 2), ( 3,0), (4, 6)} The domain is the set of first numbers x-values of the ordered pairs. The range is the set of second numbers y-values of the ordered pairs. Domain = { 3, 1, 2, 4} Range = { 2, 0, 3, 5, 6} A function is a special type of relation that pairs each domain value with exactly one range value. A function has no repeated x-values in the ordered pairs. Example: Is the relation a function? List the domain and the range. {(1, 4), (2, 5), (4, 11), (5, 13), (6, 16)} Example: Is the relation a function? List the domain and the range. {( 2, 5), (0, 2), (3, 4), (5, 6), (3, 8)}

37 When an equation represents a function, we can write the equation using function notation. y = 2x 6 becomes f(x) = 2x 6 The symbol f(x) replaces the variable y. To evaluate a function means to substitute a number for the x variable in function notation. Example: For f(x) = 2x 6, find f(4). Example: For f(x) = 1 x + 4, find f(12). 3 Examples: Is the relation a function? List the domain and the range. 1. {(1, 4), (2, 5), (4, 7), (5, 2), (6, 3)} 2. {( 2, 4), (0, 2), (3, 4), (5, 5), (3, 7)} 3. {(0, 1), (1, 3), (3, 5), (5, 7)} Example: For f(x) = x 2 + 4x 8, find f(3).

38 Chapter 4-3 Tables and Graphs A graph consists of a pair of axes and ordered pairs (x, y). Quadrant II Quadrant I (x, y) x-axis Quadrant III Quadrant IV y-axis If an equation has 2 variables its solutions will be ordered pairs (x, y). Example: Show (2, 5) is a solution of y = 2x + 1. Example: Determine whether or not the ordered pair, (x, y), is a solution. 3x + 2y = 4 a. (1, 2) b. ( 2, 5) c. (4, 4) When all of the solutions of an equation are placed on a graph, you are graphing the equation.

39 Example: Construct a table and graph the equation. y = 2x + 1 x y = 2x + 1 y Example: Construct a table and graph the equation. y = 4 2x x y = 4 2x y

40 Chapter 5-1 Graphing Linear Equations An equation whose graph forms a straight line is called a linear equation. Linear equations can be written in standard form as Ax + By = C, where A, B, and C are numbers. Example: Determine whether or not the ordered pair, (x, y), is on the graph of the equation, 2x + y = 6 a. (1, 2) b. (0, 6) c. (2, 2) Example: Complete the table and graph the linear equation. 2x + y = 1 x 2x + y = -1 y

41 Example: Complete the table and graph the linear equation. 2x + 4y = 4 x -2x + 4y = 4 y Example: Complete the table and graph the linear equation. x 2y = 6 x x 2y = 6 y

42 Chapter 5-2 Graphing Intercepts A y-intercept is the y-coordinate of a point where a graph intersects the y-axis. An x-intercept is the x-coordinate of a point where a graph intersects the x-axis. How to Find Intercepts: To find the x-intercept, replace y with 0 and solve for x. To find the y-intercept, replace x with 0 and solve for y. Example: Find the x-intercept and the y-intercept. 3x + 5y = 30 Example: Find the x-intercept and the y-intercept. 4x + 6y = 12 Graphing Linear Equations by Using Intercepts: 1. Find the x-intercept and the y-intercept. 2. Graph the intercepts. 3. Draw the line through the intercepts.

43 Example: Graph the linear equation using intercepts. 2x 4y = 8 Example: Graph the linear equation using intercepts. 6x + 3y = 12

44 Chapter 5-3 Slope of a Line The rise is the difference in the y-values of two points on a line. (2, 5) and (4, 8) rise = 8 5 = 3 The run is the difference in the x-values of two points on a line. (2, 5) and (4, 8) run = 4 2 = 2 The slope of a line is the ratio of the rise to the run for any two points on the line. If (x 1, y 1 ) and (x 2, y 2 ) are two points on the line then the slope is given by, m = rise run = y 2 y 1 x 2 x 1 Example: Find the slope of the line through the points (2, 5) and (4, 8) Example: Find the slope of the line through the points ( 1, 6) and (3, 2)

45 Positive Slope Negative Slope Zero Slope Undefined Slope Example: Graph the line and find its slope.

46 Chapter 5-5 Slope-Intercept Equation The slope-intercept linear equation is y = mx + b where m is the slope and b is the y-intercept. How to Write a Linear Equation in Slope-Intercept Form: 1. Substitute the slope m into the equation. 2. Substitute the y-intercept b into the equation. Example: Write the linear equation in slope-intercept form. m = 2 and b = 6 Example: Write the linear equation in slope-intercept form. m = 2 3 and b = 5 Example: Write the linear equation in slope-intercept form. m = 1 2 and b = 0 How to Graph a Linear Equation in Slope-Intercept Form: 1. Put a point on the y-axis for the y-intercept, b. 2. Use the slope m, rise and run, to put several more points on the graph of the linear equation. 3. Draw the line through the points.

47 Example: Graph the linear equation, y = 1 x Example: Graph the linear equation, 4x + 3y = 24.

48 Examples: Write the linear equation in slope-intercept form. 1. m = 2 and b = 1 2. m = 0 and b = 5 3. m = 3 5 and b = 7 Example: Graph the linear equation. y = 2x 3

49 Chapter 5-6 Point-Slope Form The equation of a horizontal line is written, y = b where b is the y-intercept. The equation of a vertical line is written, x = a where a is the x-intercept. Example: Write the equation of the horizontal line through ( 2, 6 ). Example: Write the equation of the vertical line through ( 4, 9 ). Point-Slope Form: The line with slope m that contains the point (x1, y1) is given by the equation y y 1 = m(x x 1 ) Example: Write the linear equation in point-slope form with m = 3 that contains (5, 7).

50 Example: Write the linear equation in point-slope form and slope-intercept form with m = 2 that contains 3 ( 3, 2 ). Example: Write the linear equation in point-slope form through ( 0, 5 ) and ( 4, 7 ). Example: Write the linear equation in point-slope form and slope-intercept form through (6, 9) and (7, 8 ). Examples: Write the linear equation in point-slope form; 1. with m = 1 2 that contains ( 4, 6 ). 2. with m = 3 4 that contains ( 8, 1 ). 3. through ( 1, 2 ) and ( 3, 10 ). 4. through ( 3, 2 ) and ( 1, 4 ).

51 Chapter 5-7 Slopes of Parallel Lines and Perpendicular Lines Two non-vertical lines are parallel if and only if they have the same slope. m 1 = m 2 Example: Write the linear equation in point-slope form of the line parallel to y = 2x + 3 through (1, 5). Example: Write the linear equation in point-slope form and slope-intercept form of the line parallel to y = 3x + 5 through ( 2, 6). Two non-vertical lines are perpendicular if and only if their slopes are negative reciprocals. m 1 = 1 m 2

52 Example: Write the linear equation in point-slope form of the line perpendicular to y = 3x 1 through (3, 2). Example: Write the linear equation in point-slope form and slope-intercept form of the line perpendicular to y = 1 2 x + 4 through ( 1, 5). Examples: 1. Write the linear equation in point-slope form of the line parallel to y = 3x + 8 through (4, 10). 2. Write the linear equation in point-slope form of the line perpendicular to y = 2x 1 through (5, 3). 3. Write the linear equation in point-slope form and slopeintercept form of the line parallel to y = 5x 7 through (1, 6). 4. Write the linear equation in point-slope form and slopeintercept form of the line perpendicular to y = 2 3 x 5 through ( 2, 4).

53 Chapter 6-1 Graphing Systems of Equations A system of equations is a set of two or more equations containing two or more variables. A solution to a system of two equations with two variables is an ordered pair (x, y) that satisfies both equations. Example: Show (4, 1) is a solution to the system of equations, x + 2y = 6 x y = 3 To find a solution to a system of equations, graph each equation and find the point of intersection. Use a graph to solve the system of equations: Example: y = 2x + 1 y = 2x + 5

54 Use a graph to solve the system of equations: Example: y = 1 2 x 1 y = 1 2 x + 3 Use a graph to solve the system of equations: Example: y = 3 4 x 1 y = 1 4 x + 3

55 Example: Is (5, 2) is a solution to the system of equations? 2x + y = 12 x 3y = 11 Use a graph to solve the system of equations: Example: y = 2 3 x 1 y = 1 3 x + 2

56 Chapter 6-2 Substitution Method The substitution property can be used to reduce a system of two equations with two variables to one equation with one variable. To solve a system of equations by substitution: 1. Solve for a variable in one of the equations, if necessary. 2. Substitute for the same variable in the other equation. 3. Solve the resulting equation. 4. Back substitute to solve for the isolated variable. Use the substitution method to solve for x and y: Example: x = 2y x + 3y = 10 Example: 2x + 4y = 14 y = x 1

57 Example: 2x + y = 5 y x = 4 Example: 4y 5x = 9 x 4y = 11 Use the substitution method to solve for x and y: 1. 4x + 2y = x = 2y 2 y = 3x 3x 4y = x + y = x + y = 8 x + y = 7 3x + 2y = 9

58 Chapter 6-3 Elimination Method Another method for solving systems of equations is called elimination. To solve a system of equations by elimination: 1. Arrange both equations in ax + by = c form. 2. Eliminate a variable by adding like terms. 3. Solve the resulting equation. 4. Back substitute to solve for the other variable. Use the elimination method to solve for x and y: Example: x y = 3 x + y = 5 Example: 5x + 3y = 13 2x 3y = 1

59 If adding does not eliminate a variable multiply one equation by 1 then try adding. Example: 2x + 3y = 6 2x y = 2 Example: x + 2y = 19 5x + 2y = 1 Use the elimination method to solve for x and y: 1. x + 2y = 4 x y = x 6y = 10 2x 5y = 9

60 Ch 6-3 Elimination Method with Multiplication If elimination does not reduce a system to one equation with one variable, then multiply one or both equations by constants to eliminate one of the variables. To solve a system by elimination with multiplication: 1. Arrange both equations in ax + by = c form. 2. Choose a variable to eliminate. 3. Multiply one or both equations by constants so that the variable to be eliminated has opposite coefficients. 4. Eliminate that variable by adding. 5. Solve the resulting equation. 6. Back substitute to solve for the other variable. Use the elimination with multiplication method to solve: Example: 5x + 2y = 30 3x y = 4 Example: 2x + y = 3 x + 4y = 6

61 Example: 3x 5y = 4 4x + 2y = 12 Example: 3x + 2y = 12 2x + 5y = 8 Example: 5x + 2y = 7 3x + 7y = 10 Use the elimination with multiplication method to solve: 1. 2x + y = x + 5y = 26 x + 3y = 12 3x 4y = 25

62 Chapter 6-4 Special Systems of Equations When a system of equations has at least one solution it is called a consistent system. If it has exactly one solution it is called independent. If it has more than one solution it is called dependent. When a system of equations has no solution it is called an inconsistent system. To solve a special system of equations: 1. Solve the system by elimination or substitution. 2. The resulting equation may be always true (consistent) with one or infinitely many solutions. 3. The resulting equation may be never true (inconsistent) with no solution. Solve the special system of equations: Example: 7x y = 3 7x + y = 3 Example: y = 2x 2 2x + y = 1

63 Example: 5x + 3y = 8 5x 3y = 8 Example: y = 3x + 6 9x 3y = 18 Solve the special system of equations: 1. x 2y = 3 2. y = x 6 x + 2y = 3 x + y = 2 3. x = 2y x + y = 6 2x 4y = 2 6x + 2y = 4

64 Chapter 6-5 Applying Systems of Equations Systems of equations can be used to solve problems involving two or more unknown quantities. Example: The school is selling tickets to its play. Five student tickets and seven adult tickets sell for $99. Four student tickets and fourteen adult tickets sell for $180. Find the price of each ticket. Example: The school sells pizza and salad. A pizza and a salad costs $16. Ten pizzas and two salads cost $104. How much does each item cost? Example: The cash register at school has 100 coins, only nickels and dimes. The value of the nickels and dimes is $6.50, how many of each coin is in the register? Example: The classroom donation bank has 50 coins, only dimes and quarters. The value of the quarters and dimes is $7.25, how many of each coin are in the donation bank?

65 Chapter 6-6 Solving Linear Inequalities A linear inequality has the form Ax + By > C A solution to a linear inequality with two variables is an ordered pair (x, y) that makes the inequality true. How to Graph the Solutions to a Linear Inequality: 1. Solve the inequality for y, if necessary. 2. Graph the boundary line. Use a solid line for or. Use a dashed line for > or <. 3. Shade the region above the line for y > or y. Shade the region below the line for y < or y. Example: Graph the linear inequality, y > 2x 3

66 Example: Graph the linear inequality, y 1 2 x + 4 Example: Graph the linear inequality, 3x + 2y 6

67 Chapter 6-7 Graphing Systems of Inequalities A system of linear inequalities is two or more linear inequalities. The solution to a system of inequalities is the region that contains solutions to all the inequalities. How to the Graph Solutions of a System Linear Inequalities: 1. Graph the boundary line for each inequality. Use a solid line for or. Use a dashed line for > or <. 2. Lightly shade the region above each line for y > or y Lightly shade the region below each line for y < or y 4. The solutions are the region that is shaded twice. Example: Graph the system of inequalities. y x 3 y < x + 2

68 Example: Graph the system of inequalities. y 2x + 4 y < 3x 6 Example: Graph the system of inequalities. y < 1 2 x 3 y 2 3 x + 3