Sure! Let's solve the given system of equations step-by-step.
We are given the following system of equations:
1. [tex]\(7x - 4y = -8\)[/tex]
2. [tex]\(y = \frac{3}{4}x - 3\)[/tex]
### Step 1: Substitute Equation 2 into Equation 1
We know from Equation 2 that [tex]\(y\)[/tex] can be expressed as [tex]\(\frac{3}{4}x - 3\)[/tex]. We substitute this expression for [tex]\(y\)[/tex] in Equation 1:
[tex]\[ 7x - 4\left(\frac{3}{4}x - 3\right) = -8 \][/tex]
### Step 2: Simplify the Equation
Let's simplify the substituted equation:
[tex]\[ 7x - 4\left(\frac{3}{4}x - 3\right) = -8 \][/tex]
Distribute [tex]\(-4\)[/tex] inside the parentheses:
[tex]\[ 7x - 4 \cdot \frac{3}{4}x + 4 \cdot 3 = -8 \][/tex]
[tex]\[ 7x - 3x + 12 = -8 \][/tex]
Combine like terms:
[tex]\[ 4x + 12 = -8 \][/tex]
### Step 3: Solve for [tex]\(x\)[/tex]
Isolate [tex]\(x\)[/tex] by subtracting 12 from both sides of the equation:
[tex]\[ 4x = -8 - 12 \][/tex]
[tex]\[ 4x = -20 \][/tex]
Now, divide both sides by 4:
[tex]\[ x = \frac{-20}{4} \][/tex]
[tex]\[ x = -5 \][/tex]
### Step 4: Solve for [tex]\(y\)[/tex]
Use the value of [tex]\(x\)[/tex] in Equation 2 to find [tex]\(y\)[/tex]:
[tex]\[ y = \frac{3}{4}x - 3 \][/tex]
[tex]\[ y = \frac{3}{4}(-5) - 3 \][/tex]
[tex]\[ y = -\frac{15}{4} - 3 \][/tex]
Convert 3 to a fraction with the same denominator (4):
[tex]\[ y = -\frac{15}{4} - \frac{12}{4} \][/tex]
[tex]\[ y = -\frac{27}{4} \][/tex]
So, we have:
[tex]\[ y = -\frac{27}{4} \][/tex]
[tex]\[ y = -6.75 \][/tex]
### Final Answer
The solution to the system of equations is:
[tex]\[ x = -5 \][/tex]
[tex]\[ y = -6.75 \][/tex]
Thus, Shannon's system of equations has the solution [tex]\((-5, -6.75)\)[/tex].
