Solve the system of equations using substitution.

[tex]\[
\begin{array}{l}
3x + 2y = 5 \\
x = 2y + 7
\end{array}
\][/tex]

A. [tex]\((3, -2)\)[/tex]
B. [tex]\((5, -5)\)[/tex]
C. [tex]\((7, 0)\)[/tex]
D. [tex]\((11, 2)\)[/tex]



Answer :

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].