Chapter 6 Systems of Equations and Inequalities
6.1 Solve Linear Systems by Graphing I can graph and solve systems of linear equations. CC.9-12.A.CED.2, CC.9-12.A.CED.3, CC.9-12.A.REI.6
What is a system of equations? A system of equations is when you have two or more equations using the same variables. The solution to the system is the point that satisfies ALL of the equations. This point will be an ordered pair. When graphing, you will encounter three possibilities.
Intersecting Lines The point where the lines intersect is your solution. The solution of this graph is (1, 2) The system is called a consistent independent system (1,2)
Parallel Lines These lines never intersect! Since the lines never cross, there is NO SOLUTION! Parallel lines have the same slope with different y-intercepts. 2 Slope = = 2 1 y-intercept = 2 y-intercept = -1
Coinciding Lines These lines are the same! Since the lines are on top of each other, there are INFINITELY MANY SOLUTIONS! Coinciding lines have the same slope and y-intercepts. 2 Slope = = 2 1 y-intercept = -1
What is the solution of the system graphed below? 1. (2, -2) 2. (-2, 2) 3. No solution 4. Infinitely many solutions
Solving a system of equations by graphing. There are 3 steps to solving a system using a graph. Step 1: Graph both equations. Graph using slope and y intercept or x- and y-intercepts. Be sure to use a ruler and graph paper! Step 2: Do the graphs intersect? This is the solution! LABEL the solution! Step 3: Check your solution. Substitute the x and y values into both equations to verify the point is a solution to both equations.
Example: Solve the system of equations by graphing. y x 6 y 3x
Example: Solve the system of equations by graphing. x y 3 x y 1
HOMEWORK: p.372 #1, 2, 3 25 odd, 31, 33
6.2 Solve Linear Systems by Substitution I can solve systems of linear equations by substitution. CC.9-12.A.CED.2, CC.9-12.A.CED.3, CC.9-12.A.REI.5, CC.9-12.A.REI.6
What is Substitution? Substitution is when you replace one thing with another thing. Substitution is ideal to use if one of the variables has already been isolated in the original system of equations. Example: y = -2x + 3
Example: Use substitution to solve each system of equations. x 2y 2x 3y 5
Example: Use substitution to solve each system of equations. 5x y y 3x 1 3
Example: Use substitution to solve each system of equations. 3x 1 y 9x 3y 3
Example: Use substitution to solve each system of equations. -5x y = 12 3x 5y = 4
HOMEWORK: p.381 #1, 2, 3 35 odd, 36
6.3 Solve Linear Systems by Adding or Subtracting I can solve linear systems using elimination. CC.9-12.A.CED.2, CC.9-12.A.CED.3, CC.9-12.A.REI.6
Remember, adding two numbers together which have the same absolute value but are opposite in sign results in a value of zero. 1 1 1-1 -1-1 Each algebra tile has the same absolute value but is opposite in sign. Each pair of tiles makes a zero pair. This same principle can be applied to variables. Below is a zero pair for the variable, x. x -x
We can apply the idea of a zero pair to systems of equations so that one of the variables can be eliminated. Look at the following system of equations. What do you observe? x y 6 x y 2 Zero pair In each equation, there is a y. However, the signs in the two equations are opposite. If added together, the y and the -y would make a zero pair.
Add the following two equations together by adding like terms together. x x y y 6 2 2x 0 8 Now solve for x. 2x = 8 2 2 x = 4 Now pick one of the two equations and substitute the value of x into that equation and solve for y. x + y = 6 4 + y = 6-4 -4 y = 2 Write the solution as an ordered pair. Solution: (4,2)
x x y y 6 2 Solution: (4,2) To check your work, substitute the values for the variables into each equation and determine if it is true. x y 6 4 2 6 6 6 The solution is correct. x y 2 4 2 2 2 2
Addition worked when the signs of one of the variables were opposite. When variables have the same sign, instead of adding, subtract one equation from the other. Remember that subtraction is really the same as adding the opposite. Let s take a look at the following system of equations: 4x y 9 3x y 6 The signs of the variables are all positive. Therefore, in order to solve this system, we can subtract the bottom equation from the top equation.
Subtract the bottom equation from the top equation. (-) 4x y 9 3x y 6 x or x 3 0 3 Since we have the value for x. Pick one of the equations and substitute the value 3 for x and solve for y. 3x + y = 6 (3)(3) + y = 6 9 + y = 6-9 -9 y = -3 Write the solution as an ordered pair. Solution: (3,-3)
4x y 9 3x y 6 Solution: (3,-3) To check your work, substitute the values for the variables into each equation and determine if it is true. 4x y 9 3x y 6 4( 3) ( 3) 9 12 3 9 3( 3) ( 3) 6 9 3 6 9 9 The solution is correct. 6 6
Solve the following system of equations by using addition or subtraction. 2x y 2 2x y 5 This system of equations can be solved by using subtraction. (-) 2x y 2 2x y 5 0 0 7 When subtraction is used, there are no more variables (x or y) remaining and the result is an incorrect statement. We know that 0 7 Therefore, this system of equations represents parallel lines and there is no solution.
1. Arrange the two equations so that the like terms are in vertical columns. 2. If the signs of the variables are opposite, add the two equations together to eliminate one of the variables. 3. If the signs of the variables are the same, then subtract one of the equations from the other equation. 4. Solve for the remaining variable. 5. Substitute the value of the variable into one of the equations and solve for the other variable. Write as an ordered pair. 6. Check the solution by substituting the values of the two variables into each equation.
HOMEWORK: p.389 #1, 2, 3 41 odd
6.4 Solve Linear Systems by Multiplying First I can solve linear systems by multiplying first. CC.9-12.A.CED.2, CC.9-12.A.CED.3, CC.9-12.A.REI.5, CC.9-12.A.REI.6
Solving Systems of Equations So far, we have solved systems using graphing, substitution, and elimination. These notes go one step further and show how to use ELIMINATION with multiplication. What happens when the coefficients are not the same? We multiply the equations to make them the same! You ll see
Solving a system of equations by elimination using multiplication. Step 1: Put the equations in Standard Form. Standard Form: Ax + By = C Step 2: Determine which variable to eliminate. Look for variables that have the same coefficient. Step 3: Multiply the equations and solve. Step 4: Plug back in to find the other variable. Step 5: Check your solution. Solve for the variable. Substitute the value of the variable into the equation. Substitute your ordered pair into BOTH equations.
Example 1: Solve the system using elimination. 2x + 2y = 6 3x y = 5 Step 1: Put the equations in Standard Form. Step 2: Determine which variable to eliminate. They already are! None of the coefficients are the same! Find the least common multiple of each variable. LCM = 6x, LCM = 2y Which is easier to obtain? 2y (you only have to multiply the bottom equation by 2)
Example 1: Solve the system using elimination. Step 3: Multiply the equations and solve. Step 4: Plug back in to find the other variable. 2x + 2y = 6 3x y = 5 Multiply the bottom equation by 2 2x + 2y = 6 2x + 2y = (2)(3x y = 5) 6 (+) 6x 2y = 8x = 16 10 x = 2 2(2) + 2y = 6 4 + 2y = 6 2y = 2 y = 1
Example 1: Solve the system using elimination. 2x + 2y = 6 3x y = 5 Step 5: Check your solution. (2, 1) 2(2) + 2(1) = 6 3(2) - (1) = 5 Solving with multiplication adds one more step to the elimination process.
Example 2: Solve the system using elimination. x + 4y = 7 4x 3y = 9
What is the first step when solving with elimination? 1. Add or subtract the equations. 2. Multiply the equations. 3. Plug numbers into the equation. 4. Solve for a variable. 5. Check your answer. 6. Determine which variable to eliminate. 7. Put the equations in standard form.
Which variable is easier to eliminate? 1. x 2. y 3. 6 4. 4 3x + y = 4 4x + 4y = 6
Example 3: Solve the system using elimination. 3x + 4y = -1 4x 3y = 7
What is the best number to multiply the top equation by to eliminate the x s? 1. -4 2. -2 3. 2 4. 4 3x + y = 4 6x + 4y = 6
Solve using elimination. 2x 3y = 1 x + 2y = -3 1. (2, 1) 2. (1, -2) 3. (5, 3) 4. (-1, -1)
Find two numbers whose sum is 18 and whose difference 22. 1. 14 and 4 2. 20 and -2 3. 24 and -6 4. 30 and 8
HOMEWORK: p.396 #1, 2, 3 33 odd, 37-39
6.5 Solve Special Types of Linear Systems I can identify the number of solutions of a linear system. CC.9-12.A.CED.2, CC.9-12.A.CED.3, CC.9-12.REI.5, CC.9-12.A.REI.6
What is a system of equations? A system of equations is when you have two or more equations using the same variables. The solution to the system is the point that satisfies ALL of the equations. This point will be an ordered pair. When graphing, you will encounter three possibilities.
Intersecting Lines The point where the lines intersect is your solution. The solution of this graph is (1, 2) The type is system is consistent and independent (1,2)
Parallel Lines These lines never intersect! Since the lines never cross, there is NO SOLUTION! Parallel lines have the same slope with different y-intercepts. The system is inconsistent. 2 Slope = = 2 1 y-intercept = 2 y-intercept = -1
Coinciding Lines These lines are the same! Since the lines are on top of each other, there are INFINITELY MANY SOLUTIONS! Coinciding lines have the same slope and y-intercepts. The system is consistent and dependent. 2 Slope = = 2 1 y-intercept = -1
Example: State whether each system is consistent and independent, consistent and dependent, or inconsistent.
Example: State whether each system is consistent and independent, consistent and dependent, or inconsistent.
Example: The system of equations below represents the tracks of two trains. Do the tracks intersect, run parallel, or are the trains running on the same track? 4x 2y 6 8x 4y 20
Example: Show that the linear system has infinitely many solutions. 4x2y 8 y 2x4
HOMEWORK: p.406 #1, 2, 3 35odd, 37, 38
6.6 Solve Systems of Linear Inequalities I can solve systems of linear inequalities in two variables. CC.9-12..A.CED.2, CC.9-12.A.CED.3, CC.9-12.A.REI.12
What is a system of inequalities? Systems of inequalities are similar to systems of equations. A solution is still an ordered pair that is true in both statements. The graphing is a little more sophisticated - it s all about the shading.
Graphing Method Example: Graph the inequalities on the same plane: x + y < 6 and 2x - y > 4. Before we graph them simultaneously, let s look at them separately. y 10 x -10 10 Graph of x + y < 6. ---> -10
Graphing Method This is: 2x - y > 4. 10 y x -10 10 So what happens when we graph both inequalities simultaneously? -10
Coolness Discovered! Wow! The solution to the system is the brown region - where the two shaded areas coincide. The green region and red regions are outside the solution set. y 10 x -10 10-10
So what were the steps? Graph first inequality Shade lightly (or use colored pencils) Graph second inequality Shade lightly (or use colored pencils) Shade darkly over the common region of intersection. That is your solution!
Example: Solve each system of inequalities by graphing. If the system does not have a solution, write no solution. x 3 y x 4
Example: Solve each system of inequalities by graphing. If the system does not have a solution, write no solution. x y 2 2y 2x 2
HOMEWORK: p.417 #1, 2, 3 39 odd