Answer :
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]
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]
