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]
