Questions: Solve the given system by the substitution method. 2x + y = 11 7x - 3y = 19

Solve the given system by the substitution method. 2x + y = 11 7x - 3y = 19
Transcript text: Solve the given system by the substitution method. \[ \left\{\begin{array}{l} 2 x+y=11 \\ 7 x-3 y=19 \end{array}\right. \]
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Solution

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Solution Steps

To solve the given system of equations using the substitution method, follow these steps:

  1. Solve one of the equations for one variable in terms of the other variable.
  2. Substitute this expression into the other equation to solve for the second variable.
  3. Substitute the value of the second variable back into the first equation to find the value of the first variable.
Step 1: Solve the first equation for \( y \)

Given the system of equations: \[ \left\{\begin{array}{l} 2x + y = 11 \\ 7x - 3y = 19 \end{array}\right. \]

First, solve the equation \( 2x + y = 11 \) for \( y \): \[ y = 11 - 2x \]

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

Substitute \( y = 11 - 2x \) into the second equation \( 7x - 3y = 19 \): \[ 7x - 3(11 - 2x) = 19 \]

Step 3: Simplify and solve for \( x \)

Simplify the equation: \[ 7x - 33 + 6x = 19 \] Combine like terms: \[ 13x - 33 = 19 \] Add 33 to both sides: \[ 13x = 52 \] Divide by 13: \[ x = 4 \]

Step 4: Substitute \( x \) back into the expression for \( y \)

Substitute \( x = 4 \) back into \( y = 11 - 2x \): \[ y = 11 - 2(4) = 11 - 8 = 3 \]

Final Answer

\(\boxed{(x, y) = (4, 3)}\)

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