Solve the system of equations by substitution.

Question

left brace Start 2 By 1 Matrix 1st Row 1st Column 3 x minus 4 y equals 11 2nd Row 1st Column 7 x plus 3 y equals 1 EndMatrix

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Part 1
The solution of the system is xequals
  
enter your response here and yequals
  
enter your response here.
​(Type integers or simplified​ fractions.)

Answer 1

To solve the system of equations using substitution, we first write down the equations:

1. \( 3x - 4y = 11 \) (Equation 1)
2. \( 7x + 3y = 1 \) (Equation 2)

Step 1: Solve one of the equations for one variable

Let's solve Equation 1 for \( y \):

\[ 3x - 4y = 11 \]

Rearranging gives:

\[ -4y = 11 - 3x \]

Dividing by \(-4\):

\[ y = \frac{3x - 11}{4} \quad \text{(Equation 3)} \]

Step 2: Substitute \( y \) into the other equation

Now, we substitute Equation 3 into Equation 2:

\[ 7x + 3\left(\frac{3x - 11}{4}\right) = 1 \]

Multiply through by 4 to eliminate the fraction:

\[ 28x + 3(3x - 11) = 4 \]

Distributing gives:

\[ 28x + 9x - 33 = 4 \]

Combining like terms:

\[ 37x - 33 = 4 \]

Adding 33 to both sides:

\[ 37x = 37 \]

Dividing by 37:

\[ x = 1 \]

Step 3: Substitute \( x \) back into Equation 3 to find \( y \)

Now that we have \( x \), we can find \( y \) by substituting \( x = 1 \) into Equation 3:

\[ y = \frac{3(1) - 11}{4} = \frac{3 - 11}{4} = \frac{-8}{4} = -2 \]

Final solution

Thus, the solution to the system of equations is:

\[ x = 1 \quad \text{and} \quad y = -2 \]

You can write the final answer as:

The solution of the system is \( x = 1 \) and \( y = -2 \).