Answer :
To solve the system of equations
[tex]\[ \begin{cases} 7x - 4y = -8 \\ y = \frac{3}{4}x - 3 \end{cases} \][/tex]
we follow these steps:
### 1. Substitute [tex]\( y \)[/tex] from the second equation into the first equation.
Given the second equation:
[tex]\[ y = \frac{3}{4}x - 3 \][/tex]
Substitute this expression for [tex]\( y \)[/tex] in the first equation:
[tex]\[ 7x - 4\left(\frac{3}{4}x - 3\right) = -8 \][/tex]
### 2. Simplify and solve for [tex]\( x \)[/tex].
First, distribute the [tex]\(-4\)[/tex]:
[tex]\[ 7x - 4 \left(\frac{3}{4}x\right) + 4 \cdot 3 = -8 \][/tex]
[tex]\[ 7x - 3x + 12 = -8 \][/tex]
[tex]\[ 4x + 12 = -8 \][/tex]
Subtract 12 from both sides:
[tex]\[ 4x = -8 - 12 \][/tex]
[tex]\[ 4x = -20 \][/tex]
Divide by 4:
[tex]\[ x = -5 \][/tex]
### 3. Substitute [tex]\( x = -5 \)[/tex] back into the second equation to find [tex]\( y \)[/tex].
Using [tex]\( y = \frac{3}{4}x - 3 \)[/tex], substitute [tex]\( x = -5 \)[/tex]:
[tex]\[ y = \frac{3}{4}(-5) - 3 \][/tex]
[tex]\[ y = -\frac{15}{4} - 3 \][/tex]
[tex]\[ y = -\frac{15}{4} - \frac{12}{4} \][/tex]
[tex]\[ y = -\frac{27}{4} \][/tex]
[tex]\[ y = -6.75 \][/tex]
### 4. Identify the closest approximate solution.
This means the solution to the system of equations is:
[tex]\[ (x, y) = (-5, -6.75) \][/tex]
Among the given options, we need to find the closest approximation to [tex]\( -6.75 \)[/tex]:
- [tex]\((-5, -7.2)\)[/tex]
- [tex]\((-5, -6.8)\)[/tex]
- [tex]\((-5, -6.2)\)[/tex]
- [tex]\((-5, -5.9)\)[/tex]
The value [tex]\( -6.75 \)[/tex] is closest to [tex]\( -6.8 \)[/tex].
### Conclusion:
The closest approximate solution to the system of equations is:
[tex]\[ \boxed{(-5, -6.8)} \][/tex]
[tex]\[ \begin{cases} 7x - 4y = -8 \\ y = \frac{3}{4}x - 3 \end{cases} \][/tex]
we follow these steps:
### 1. Substitute [tex]\( y \)[/tex] from the second equation into the first equation.
Given the second equation:
[tex]\[ y = \frac{3}{4}x - 3 \][/tex]
Substitute this expression for [tex]\( y \)[/tex] in the first equation:
[tex]\[ 7x - 4\left(\frac{3}{4}x - 3\right) = -8 \][/tex]
### 2. Simplify and solve for [tex]\( x \)[/tex].
First, distribute the [tex]\(-4\)[/tex]:
[tex]\[ 7x - 4 \left(\frac{3}{4}x\right) + 4 \cdot 3 = -8 \][/tex]
[tex]\[ 7x - 3x + 12 = -8 \][/tex]
[tex]\[ 4x + 12 = -8 \][/tex]
Subtract 12 from both sides:
[tex]\[ 4x = -8 - 12 \][/tex]
[tex]\[ 4x = -20 \][/tex]
Divide by 4:
[tex]\[ x = -5 \][/tex]
### 3. Substitute [tex]\( x = -5 \)[/tex] back into the second equation to find [tex]\( y \)[/tex].
Using [tex]\( y = \frac{3}{4}x - 3 \)[/tex], substitute [tex]\( x = -5 \)[/tex]:
[tex]\[ y = \frac{3}{4}(-5) - 3 \][/tex]
[tex]\[ y = -\frac{15}{4} - 3 \][/tex]
[tex]\[ y = -\frac{15}{4} - \frac{12}{4} \][/tex]
[tex]\[ y = -\frac{27}{4} \][/tex]
[tex]\[ y = -6.75 \][/tex]
### 4. Identify the closest approximate solution.
This means the solution to the system of equations is:
[tex]\[ (x, y) = (-5, -6.75) \][/tex]
Among the given options, we need to find the closest approximation to [tex]\( -6.75 \)[/tex]:
- [tex]\((-5, -7.2)\)[/tex]
- [tex]\((-5, -6.8)\)[/tex]
- [tex]\((-5, -6.2)\)[/tex]
- [tex]\((-5, -5.9)\)[/tex]
The value [tex]\( -6.75 \)[/tex] is closest to [tex]\( -6.8 \)[/tex].
### Conclusion:
The closest approximate solution to the system of equations is:
[tex]\[ \boxed{(-5, -6.8)} \][/tex]
