To solve the system of equations using substitution, we have the following two equations:
- \( y = 2x + 3 \) (Equation 1)
- \( 3x - y = -5 \) (Equation 2)
Step 1: Substitute Equation 1 into Equation 2
Since we have \( y \) expressed in terms of \( x \) from Equation 1, we can substitute \( y \) in Equation 2.
Substituting \( y \) in Equation 2 gives us:
\[ 3x - (2x + 3) = -5 \]
Step 2: Simplify the equation
Now, simplify the equation:
\[ 3x - 2x - 3 = -5 \]
This simplifies to:
\[ x - 3 = -5 \]
Step 3: Solve for \( x \)
Next, we add 3 to both sides of the equation to isolate \( x \):
\[ x = -5 + 3 \] \[ x = -2 \]
Step 4: Substitute \( x \) back into Equation 1 to find \( y \)
Now that we have \( x \), we can substitute \( x = -2 \) back into Equation 1 to find \( y \):
\[ y = 2(-2) + 3 \] \[ y = -4 + 3 \] \[ y = -1 \]
Step 5: Write the solution
The solution to the system of equations is:
\[ (x, y) = (-2, -1) \]
Step 6: Verify the solution
Finally, we can verify the solution by substituting \( x = -2 \) and \( y = -1 \) back into the original equations to ensure they hold true.
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For Equation 1: \[ y = 2x + 3 \Rightarrow -1 = 2(-2) + 3 \Rightarrow -1 = -4 + 3 \Rightarrow -1 = -1 \quad \text{(True)} \]
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For Equation 2: \[ 3x - y = -5 \Rightarrow 3(-2) - (-1) = -5 \Rightarrow -6 + 1 = -5 \Rightarrow -5 = -5 \quad \text{(True)} \]
Both equations verify correctly, confirming that the solution \((-2, -1)\) is indeed correct.
Thus, the final answer is:
\[ (x, y) = (-2, -1) \]