Questions: Solve by substitution. -3x + y = 19 2x - 3y = -1

Solve by substitution. -3x + y = 19 2x - 3y = -1
Transcript text: Solve by substitution. \[ \left\{\begin{array}{r} -3 x+y=19 \\ 2 x-3 y=-1 \end{array}\right. \]
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Solution

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Solution Steps

To solve the system of equations by substitution, we can follow these steps:

  1. Solve one of the equations for one variable in terms of the other.
  2. Substitute this expression into the other equation to find the value of one variable.
  3. Substitute the value found back into the first equation to find the value of the other variable.
Step 1: Solve one equation for one variable

First, we solve the first equation for \( y \): \[ -3x + y = 19 \implies y = 3x + 19 \]

Step 2: Substitute the expression into the second equation

Next, we substitute \( y = 3x + 19 \) into the second equation: \[ 2x - 3(3x + 19) = -1 \] Simplify and solve for \( x \): \[ 2x - 9x - 57 = -1 \implies -7x - 57 = -1 \implies -7x = 56 \implies x = -8 \]

Step 3: Substitute the value of \( x \) back into the expression for \( y \)

Now, substitute \( x = -8 \) back into the expression \( y = 3x + 19 \): \[ y = 3(-8) + 19 = -24 + 19 = -5 \]

Final Answer

\(\boxed{(x, y) = (-8, -5)}\)

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