Question

The linear combination method gives a solution of (–4, 2) for which of these systems of linear equations? 3 x 13 y = 14. 6 x 11 y = negative 2. 4 x 5 y = 12. 8 x 3 y = negative 4. 5 x 4 y = 12. 7 x 8 y = 12. 10 x 3 y = 8. 17 x 6 y = 10.

Answer 1

(a) 3x +13y = 14; 6x +11y = -2

Explanation:

Regardless of the solution method, a given point will only be a solution to a system of equations if it satisfies all of the equations in the system.

We can check to see which equations include the point (-4, 2) as one of their solutions. In each case, we will substitute x=-4 and y=2 to see if the equation is true.

A) 3x +13y = 14; 6x +11y = -2

3(-4) +13(2) = -12 +26 = 14 . . . true

6(-4) +11(2) = -24 +22 = -2 . . . true

(-4, 2) is a solution to the system 3x +13y = 14; 6x +11y = -2.

B) 4x +5y = 12; 8x +3y = -4

4(-4) +5(2) = -16 +10 = -6 ≠ 12 . . . not a solution

C) 5x +4y = 12; 7x +8y = 12

5(-4) +4(2) = -20 +8 = -12 ≠ 12 . . . not a solution

D) 10x +3y = 8; 17x +6y = 10

10(-4) +3(2) = -40 +6 = -34 ≠ 8 . . . not a solution

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Additional comment

The "linear combination method" requires we combine the equations resulting from multiplying one or both by factors designed to eliminate one of the variables in the combined equation.

Here, we see that the coefficients of x in the first system of equations are related by a factor of 2, so we can eliminate the x terms by subtracting the second equation from twice the first equation. Solving the first system by the linear combination method, we get ...

2(3x +13y) -(6x +11y) = 2(14) -(-2)

6x +26y -6x -11y = 28 +2 . . . . . . eliminate parentheses

15y = 30 . . . . . collect terms

y = 2 . . . . . divide by 15

3x +13(2) = 14 . . . . . substitute for y in the first equation

3x = -12 . . . . . . . subtract 26

x = -4 . . . . . . divide by 3

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A graphing calculator can help identify equations for which (-4, 2) is a solution.