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Which of the following equations shows how substitution can be used to solve the following system of equations?

[tex]\[
\begin{cases}
y = 2x - 7 \\
3x + 4y = 16
\end{cases}
\][/tex]

A. [tex]\(3(2x - 7) + 4y = 16\)[/tex]

B. [tex]\(3x + 4y = 2x - 7\)[/tex]

C. [tex]\(3x + 4(2x - 7) = 16\)[/tex]


Sagot :

To solve the system of equations using substitution, we start with the given system:

[tex]\[ \begin{cases} y = 2x - 7 \\ 3x + 4y = 16 \end{cases} \][/tex]

The substitution method involves replacing one variable with an equivalent expression in terms of the other variable. Here, the first equation [tex]\( y = 2x - 7 \)[/tex] allows us to express [tex]\( y \)[/tex] in terms of [tex]\( x \)[/tex]. We can then substitute this expression for [tex]\( y \)[/tex] into the second equation.

Substitute [tex]\( y = 2x - 7 \)[/tex] into the second equation [tex]\(3x + 4y = 16\)[/tex]:

[tex]\[ 3x + 4(2x - 7) = 16 \][/tex]

This substitution involves replacing [tex]\( y \)[/tex] in the second equation with [tex]\( 2x - 7 \)[/tex], which simplifies directly from the first equation. Therefore, the correct equation that shows how substitution can be used is:

[tex]\[ 3x + 4(2x - 7) = 16 \][/tex]

To confirm, let's solve this substitution step-by-step:

1. Substitute [tex]\( y \)[/tex] in the second equation:
[tex]\[ 3x + 4(2x - 7) = 16 \][/tex]

2. Distribute the [tex]\( 4 \)[/tex] inside the parentheses:
[tex]\[ 3x + 8x - 28 = 16 \][/tex]

3. Combine like terms:
[tex]\[ 11x - 28 = 16 \][/tex]

4. Add 28 to both sides to get:
[tex]\[ 11x = 44 \][/tex]

5. Divide by 11 to solve for [tex]\( x \)[/tex]:
[tex]\[ x = 4 \][/tex]

6. Substitute [tex]\( x = 4 \)[/tex] back into [tex]\( y = 2x - 7 \)[/tex] to find [tex]\( y \)[/tex]:
[tex]\[ y = 2(4) - 7 = 8 - 7 = 1 \][/tex]

The solution to the system is [tex]\( x = 4 \)[/tex] and [tex]\( y = 1 \)[/tex].

Thus, the correct substitution equation is:

[tex]\[ 3x + 4(2x - 7) = 16 \][/tex]