To solve the given system of equations using substitution, follow these steps:
### 1. Write the system of equations
We are provided with the following system of equations:
[tex]\[
\begin{array}{l}
3x + 2y = 5 \quad \text{(Equation 1)}\\
x = 2y + 7 \quad \text{(Equation 2)}
\end{array}
\][/tex]
### 2. Substitute Equation 2 into Equation 1
From Equation 2, we know that [tex]\( x = 2y + 7 \)[/tex]. Substitute this expression for [tex]\( x \)[/tex] into Equation 1:
[tex]\[
3(2y + 7) + 2y = 5
\][/tex]
### 3. Simplify the resulting equation
Expand and simplify the equation:
[tex]\[
3(2y + 7) + 2y = 5
\][/tex]
[tex]\[
6y + 21 + 2y = 5
\][/tex]
Combine like terms:
[tex]\[
8y + 21 = 5
\][/tex]
### 4. Solve for y
Isolate [tex]\( y \)[/tex]:
[tex]\[
8y + 21 = 5
\][/tex]
Subtract 21 from both sides:
[tex]\[
8y = 5 - 21
\][/tex]
[tex]\[
8y = -16
\][/tex]
Divide both sides by 8:
[tex]\[
y = \frac{-16}{8}
\][/tex]
[tex]\[
y = -2
\][/tex]
### 5. Substitute y back into Equation 2 to solve for x
Now, substitute [tex]\( y = -2 \)[/tex] back into Equation 2:
[tex]\[
x = 2(-2) + 7
\][/tex]
[tex]\[
x = -4 + 7
\][/tex]
[tex]\[
x = 3
\][/tex]
### 6. Validate the solution
Our solution is [tex]\( x = 3 \)[/tex] and [tex]\( y = -2 \)[/tex]. The coordinates (3, -2) should satisfy both equations.
Check Equation 1:
[tex]\[
3x + 2y = 5
\][/tex]
Substitute [tex]\( x = 3 \)[/tex] and [tex]\( y = -2 \)[/tex]:
[tex]\[
3(3) + 2(-2) = 5
\][/tex]
[tex]\[
9 - 4 = 5
\][/tex]
The equation holds true.
Check Equation 2:
[tex]\[
x = 2y + 7
\][/tex]
Substitute [tex]\( y = -2 \)[/tex]:
[tex]\[
x = 2(-2) + 7
\][/tex]
[tex]\[
x = -4 + 7
\][/tex]
[tex]\[
x = 3
\][/tex]
The equation holds true.
### 7. Identify the correct point from the given options
The point (3, -2) is the solution. Now, we match this with the given points:
- (3, -2)
- (5, -5)
- (7, 0)
- (11, 2)
The correct point is [tex]\( (3, -2) \)[/tex].
### Final solution
The solution to the system of equations is [tex]\( x = 3 \)[/tex] and [tex]\( y = -2 \)[/tex], which corresponds to the point [tex]\( (3, -2) \)[/tex].
