2x + 5 = x = x = 4
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1 98 CHAPTER 3 Algebra Textbook Reference Section LINEAR EQUATIONS AND INEQUALITIES Student CD Section.5 CLAST OBJECTIVES Solve linear equations and inequalities Solve a system of two linear equations in two unknowns To solve an equation means to determine the value of the variable that makes the equation true. To solve a linear equation, isolate the variable on one side of the equal sign. Items in an equation can be moved to the other side by performing opposite operations to both sides of the equation. a) Solve. + 5 = = = 8 x = 4 Get the variable on one side by itself. 1. The opposite of adding 5 is subtracting 5. Subtract 5 from both sides of the equation.. Now get rid of the, the coefficient of x. The opposite of multiplying by is dividing by. Divide both sides of the equation by. 3. The value of the variable x is We can check this by replacing x with 4 in the original equation. (4) + 5 = = 13 (True) b) Solve. 3(x ) + 4 = 5[10 3(x + 4)] 3(x ) + 4 = 5[10 3(x + 4)] 3x = 5[10 3x 1] 3x = 5[-3x ] 3x = -15x x +15x 18x = = + 18x = x = = 18 9 Note: Solve the equation, Determine the solution, and Find the roots all mean the same thing. Linear Inequalities Inequalities carry inequality signs: >, <,,. Linear inequalities are solved similarly to solving equations. When multiplying or dividing by a negative quantity across an inequality sign, you must reverse the inequality sign.
2 SECTION 3.3 Linear Equations and Inequalities 99 c) Solve. 1 t 6 15t t 6 15t t -15t -3t t 15-3t t -5 Get the variable on one side by itself. 1. The opposite of adding 15t is subtracting 15t. Subtract 15t from both sides of the inequality.. The opposite of subtracting 6 is adding 6. Add 6 to both sides of the inequality. 3. Now get rid of -3, the coefficient of t by dividing both sides of the inequality by -3. Since we are dividing both sides of the inequality by a negative number, we must reverse the inequality sign from less than or equal to greater than or equal. d) Solve. 3(x + 5) > (x 1) 3(x + 5) > (x 1) 3x + 15 > x + 1 3x + 15 > x + 3 +x + x + 15 > > x > -3 Check Your Progress 3.3 Solve each of the following x = 3x t + 14 = 3t y + 3 = ( 4y 1) 4. ( a + 3) = 4[ a ( 5 + a) ] 3 5. x + 3 x = ( x ) x > y 1 y x x + 1 > 7x 30 x 10. 4( x 1) + 5x < 7x 4
3 100 CHAPTER 3 Algebra System of Equations: Elimination Method (Two Equations & Two Unknowns) The idea is to add the equations with the intent that one of the variables will be eliminated, leaving a simple, one-variable equation to solve. Sometimes it may be necessary to multiply the equation(s) by a constant before adding them in order to eliminate one of the variables. The coefficients of the variable to be eliminated should be opposites. e) Solve the following system. x y = 3 x y = 3 = 10 x = 5 Now use either of the equations in the system to solve for y. 5 + y = y = 1. Add the equations to eliminate the variable y.. Follow the process for solving a linear equation. 3. Pick either equation and replace the variable x with Follow the process for solving a linear equation. Answer: x = 5, y =, also written as (5, ) Check. 5 + = 7 (True) 5 = 3 (True) f) Solve. + y y = 8 + y y = 8 + y y = 16 0 = 0 x can be any real number. + y ( y) = (8) 1. Multiply the second equation by in an effort to eliminate the variable x.. Now, add both equations. 3. Note that 0 = 0 is a true statement. Thus, there are infinitely many solutions to this system. To determine the values of y, solve either equation for y. x y = y = y = 8 Answer: (x, y) where x can be any real number and y = 8.
4 SECTION 3.3 Linear Equations and Inequalities 101 Example g) Solve the system. = 1 = 1 = 1 6y = 1 0 = 4 (False) Answer: No solution. Solution = 1 ( ) = ( 6) 1. Multiply the second equation by - in an effort to eliminate the variable x.. Now, add both equations. 3. Note that 0 = 4 is not a true statement. Thus, there is no solution to this system. Check Your Progress 3.3 Solve the following systems. x y = 11. x + y = 4 1. y = 7 3x + 4y 13. 3x y = 8 = x y = 1 10x + 4y = y = y = y = 8x 6y = 4
5 10 CHAPTER 3 Algebra See If You Remember SECTIONS 3.1 & Simplify: 4 π 3 + π. Simplify: Simplify: Simplify: Write 6,190,000 in scientific notation. 6. Write 0.39 in scientific notation. 7. Write in standard form. 8. Write the result in scientific notation: ( ) ( ) 9. Write the result in scientific notation: Name the property: x + y = y + x 11. Name the property: ( y ) 1 x y x + = + 1. Name the property: 6 x + y = ( 3x + y)
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