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3. Lesson 7-3 Elimination Using Addition and Subtraction
Lesson 7-4 Elimination Using Multiplication
Contents

4. Example 1 Elimination Using Addition
Example 2 Write and Solve a System of Equations
Example 3 Elimination Using Subtraction
Example 4 Elimination Using Subtraction
Lesson 3 Contents

5. Use elimination to solve the system of equations.
Since the coefficients of the x terms, –3 and 3, are additive inverses, you can eliminate the x terms by adding the equations.
Write the equation in column form and add.
Notice that the x value is eliminated.
Divide each side by –2.
Simplify.
Example 3-1a

6. Now substitute –15 for y in either equation to find the value of x.
First equation
Replace y with –15.
Simplify.
Add 60 to each side.
Simplify.
Divide each side by –3.
Simplify.
Answer: The solution is (–24, –15).
Example 3-1a

7. Use elimination to solve the system of equations.
Answer: (2, 1)
Example 3-1b

8. Let x represent the first number and y represent the second number.
Four times one number minus three times another number is 12. Two times the first number added to three times the second number is 6. Find the numbers.
Let x represent the first number and y represent the second number.
Four times one number
minus
three times another number is
12. 4x – 3y = 12
Two times the first number
added to
three times the second number is 6. 2x + 3y = 6
Example 3-2a

9. Use elimination to solve the system.
Write the equation in column form and add.
Notice that the y value is eliminated.
Divide each side by 6.
Simplify.
Example 3-2a

10. Now substitute 3 for x in either equation to find the value of y.
First equation
Replace x with 3.
Simplify.
Subtract 12 from each side.
Simplify.
Divide each side by –3.
Simplify.
Answer: The numbers are 3 and 0.
Example 3-2a

11. Four times one number added to another number is –10
Four times one number added to another number is –10. Three times the first number minus the second number is –11. Find the numbers.
Answer: –3, 2
Example 3-2b

12. Use elimination to solve the system of equations.
Since the coefficients of the x terms, 4 and 4, are the same, you can eliminate the x terms by subtracting the equations.
Write the equation in column form and subtract.
Notice that the x value is eliminated.
Divide each side by 5.
Simplify.
Example 3-3a` `

13. Now substitute 2 for y in either equation to find the value of x.
Second equation
Simplify.
Add 6 to each side.
Simplify.
Divide each side by 4.
Simplify.
Answer: The solution is (6, 2).
Example 3-3a` `

14. Use elimination to solve the system of equations.
Answer: The solution is (2, –6).
Example 3-3b

15. Multiple-Choice Test Item
If and what is the value of y? A (3, –8) B 3 C –8 D (–8, 3)
Read the Test Item
You are given a system of equations, and you are asked to find the value of y.
Example 3-4a

16. Write the equation in column form and subtract.
Solve the Test Item
You can eliminate the y terms by subtracting one equation from the other.
Write the equation in column form and subtract.
Notice that the y value is eliminated.
Divide each side by 14.
Simplify.
Example 3-4a

17. Now substitute 3 for x in either equation to solve for y.
First equation
Simplify.
Subtract 24 from each side.
Simplify.
Notice that B is the value of x and A is the solution of the system of equations. However, the question asks for the value of y.
Answer: C
Example 3-4a

18. Multiple-Choice Test Item
If and what is the value of x? A 4 B (4, –4) C (–4, 4) D –4
Answer: D
Example 3-4b

19. End of Lesson 3

20. Example 1 Multiply One Equation to Eliminate
Example 2 Multiply Both Equations to Eliminate
Example 3 Determine the Best Method
Example 4 Write and Solve a System of Equations
Lesson 4 Contents

21. Use elimination to solve the system of equations.
Multiply the first equation by –2 so the coefficients of the y terms are additive inverses. Then add the equations.
Multiply by –2.
Add the equations.
Divide each side by –1.
Simplify.
Example 4-1a

22. Now substitute 9 for x in either equation to find the value of y.
First equation
Simplify.
Subtract 18 from each side.
Simplify.
Answer: The solution is (9, 5).
Example 4-1a

23. Use elimination to solve the system of equations.
Answer: (5, 1)
Example 4-1b

24. Use elimination to solve the system of equations.
Method 1 Eliminate x.
Multiply by 3.
Multiply by –4.
Add the equations.
Divide each side by 29.
Simplify.
Example 4-2a

25. Now substitute 4 for y in either equation to find x.
First equation
Simplify.
Subtract 12 from each side.
Simplify.
Divide each side by 4.
Simplify.
Answer: The solution is (–1, 4).
Example 4-2a

26. Method 2 Eliminate y. Multiply by 5. Multiply by 3. Add the equations.
Divide each side by 29.
Simplify.
Example 4-2a

27. Now substitute –1 for x in either equation.
First equation
Simplify.
Add 4 to each side.
Simplify.
Divide each side by 3.
Simplify.
Answer: The solution is (–1, 4), which matches the result obtained with Method 1.
Example 4-2a

28. Use elimination to solve the system of equations.
Answer: (4, –1)
Example 4-2b

29. For an exact solution, an algebraic method is best.
Determine the best method to solve the system of equations. Then solve the system.
For an exact solution, an algebraic method is best.
Since neither the coefficients for x nor the coefficients for y are the same or additive inverses, you cannot use elimination using addition or subtraction.
Since the coefficient of the x term in the first equation is 1, you can use the substitution method. You could also use the elimination method using multiplication.
Example 4-3a

30. The following solution uses substitution.
First equation
Subtract 5y from each side.
Simplify.
Example 4-3a

31. Distributive Property
Second equation
Distributive Property
Combine like terms.
Subtract 12 from each side.
Simplify.
Divide each side by –22.
Simplify.
Example 4-3a

32. Subtract 5 from each side.
First equation
Simplify.
Subtract 5 from each side.
Simplify.
Answer: The solution is (–1, 1).
Example 4-3a

33. Determine the best method to solve the system of equations
Determine the best method to solve the system of equations. Then solve the system.
Answer: The best method to use is elimination using subtraction because the coefficient of y is the same in both equations; (3, 5).
Example 4-3b

34. Transportation A fishing boat travels 10 miles downstream in 30 minutes. The return trip takes the boat 40 minutes. Find the rate of the boat in still water.
Let the rate of the boat in still water. Let the rate of the current. Use the formula rate  time distance, or Since the rate is miles per hour, write 30 minutes as hour and 40 minutes as hour.
Example 4-4a

35. 10 Upstream Downstream d t r
This system cannot easily be solved using substitution. It cannot be solved by just adding or subtracting the equations.
The best way to solve this system is to use elimination using multiplication. Since the problem asks for b, eliminate c.
Example 4-4a

36. Answer: The rate of the boat is 17.5 mph.
Multiply by .
Multiply by .
Add the equations.
Multiply each
side by
Simplify.
Answer: The rate of the boat is 17.5 mph.
Example 4-4a

37. Transportation A helicopter travels 360 miles with the wind in 3 hours
Transportation A helicopter travels 360 miles with the wind in 3 hours. Te return trip against the wind takes the helicopter 4 hours. Find the rate of the helicopter in still air.
Answer: mph
Example 4-4b

38. End of Lesson 4

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