To solve the system of equations using the elimination method, follow these steps:
Equations given:
[tex]\[
\begin{array}{l}
10x + 2y = 22 \quad \text{(1)} \\
3x - 4y = -21 \quad \text{(2)}
\end{array}
\][/tex]
Step 1: Align the equations and identify a variable to eliminate.
We have:
[tex]\[
10x + 2y = 22
\][/tex]
[tex]\[
3x - 4y = -21
\][/tex]
Step 2: Make the coefficients of [tex]\( y \)[/tex] equal in both equations by multiplying the equations as needed.
To eliminate [tex]\( y \)[/tex], we will manipulate the coefficients of [tex]\( y \)[/tex] to be opposites. Multiply the first equation by 2 to get the coefficient of [tex]\( y \)[/tex] to be 4:
[tex]\[
2(10x + 2y) = 2(22)
\][/tex]
[tex]\[
20x + 4y = 44 \quad \text{(3)}
\][/tex]
The second equation already has [tex]\( -4y \)[/tex], so we can use it directly:
[tex]\[
3x - 4y = -21 \quad \text{(2')}.
\][/tex]
Step 3: Add the two equations to eliminate [tex]\( y \)[/tex].
[tex]\[
20x + 4y + 3x - 4y = 44 - 21
\][/tex]
[tex]\[
20x + 3x = 23
\][/tex]
[tex]\[
23x = 23
\][/tex]
Step 4: Solve for [tex]\( x \)[/tex].
[tex]\[
x = \frac{23}{23}
\][/tex]
[tex]\[
x = 1
\][/tex]
Step 5: Substitute [tex]\( x = 1 \)[/tex] into one of the original equations to find [tex]\( y \)[/tex].
Substitute [tex]\( x = 1 \)[/tex] into the first equation (1):
[tex]\[
10(1) + 2y = 22
\][/tex]
[tex]\[
10 + 2y = 22
\][/tex]
[tex]\[
2y = 22 - 10
\][/tex]
[tex]\[
2y = 12
\][/tex]
[tex]\[
y = \frac{12}{2}
\][/tex]
[tex]\[
y = 6
\][/tex]
Step 6: Write the solution as an ordered pair.
The solution to the system of equations is:
[tex]\[
(x, y) = (1, 6)
\][/tex]
Step 7: Verify the ordered pair with the given options.
A. [tex]\((1, 6)\)[/tex] \
B. [tex]\((3, 8)\)[/tex] \
C. [tex]\((2, 5)\)[/tex] \
D. [tex]\((5, 4)\)[/tex]
The correct ordered pair is [tex]\((1, 6)\)[/tex], which corresponds to option A. Therefore, the correct option is:
Correct Answer: A. [tex]\((1, 6)\)[/tex]
