Answered

Using the linear combination method, what is the solution to the system of linear equations:

[tex]\[ 7x - 2y = -20 \][/tex]
[tex]\[ 9x + 4y = -6 \][/tex]

A. [tex]\((-3, 2)\)[/tex]
B. [tex]\((-2, 3)\)[/tex]
C. [tex]\((2, -3)\)[/tex]
D. [tex]\((3, -2)\)[/tex]



Answer :

GinnyAnswer
To solve the system of linear equations using the linear combination method, follow these detailed steps:

Given the system of equations:
[tex]\[ 7x - 2y = -20 \][/tex]
[tex]\[ 9x + 4y = -6 \][/tex]

Step 1: Align the variables and equations.
[tex]\[ 7x - 2y = -20 \quad \text{(Equation 1)} \][/tex]
[tex]\[ 9x + 4y = -6 \quad \text{(Equation 2)} \][/tex]

Step 2: Make the coefficients of one of the variables equal and opposite.
We can make the coefficients of [tex]\( y \)[/tex] in both equations the same by multiplying Equation 1 by 2:
[tex]\[ 2 \cdot (7x - 2y) = 2 \cdot (-20) \][/tex]
[tex]\[ 14x - 4y = -40 \quad \text{(Equation 3)} \][/tex]

Equation 2 is already:
[tex]\[ 9x + 4y = -6 \quad \text{(Equation 2)} \][/tex]

Step 3: Add the two equations to eliminate [tex]\( y \)[/tex].
[tex]\[ (14x - 4y) + (9x + 4y) = -40 + (-6) \][/tex]
[tex]\[ 14x + 9x - 4y + 4y = -46 \][/tex]
[tex]\[ 23x = -46 \][/tex]

Step 4: Solve for [tex]\( x \)[/tex].
[tex]\[ x = \frac{-46}{23} \][/tex]
[tex]\[ x = -2 \][/tex]

Step 5: Substitute [tex]\( x = -2 \)[/tex] back into one of the original equations to solve for [tex]\( y \)[/tex].
Using Equation 1:
[tex]\[ 7x - 2y = -20 \][/tex]
[tex]\[ 7(-2) - 2y = -20 \][/tex]
[tex]\[ -14 - 2y = -20 \][/tex]
[tex]\[ -2y = -20 + 14 \][/tex]
[tex]\[ -2y = -6 \][/tex]
[tex]\[ y = \frac{-6}{-2} \][/tex]
[tex]\[ y = 3 \][/tex]

Step 6: Write the solution as an ordered pair.
The solution to the system of equations is:
[tex]\[ \boxed{(-2, 3)} \][/tex]