Problem Analysis:
The task involves solving a system of equations using substitution and elimination methods. The given system of equations is:
1. \( 3x + y = -2 \)
2. \( x - 2y = 5 \)
We are tasked with finding the values of \( x \) and \( y \).
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Step-by-Step Solution:
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Step 1: Solve one equation for one variable.
We will solve the second equation for \( x \):
\[ x - 2y = 5 \]
Add \( 2y \) to both sides:
\[ x = 2y + 5 \]
Now, we have \( x \) expressed in terms of \( y \):
\[ x = 2y + 5 \]
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Step 2: Substitute \( x = 2y + 5 \) into the first equation.
The first equation is:
\[ 3x + y = -2 \]
Substitute \( x = 2y + 5 \) into this equation:
\[ 3(2y + 5) + y = -2 \]
Distribute the 3:
\[ 6y + 15 + y = -2 \]
Combine like terms:
\[ 7y + 15 = -2 \]
Subtract 15 from both sides:
\[ 7y = -17 \]
Divide by 7:
\[ y = -\frac{17}{7} \]
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####
Step 3: Substitute \( y = -\frac{17}{7} \) back into \( x = 2y + 5 \).
We have:
\[ x = 2y + 5 \]
Substitute \( y = -\frac{17}{7} \):
\[ x = 2\left(-\frac{17}{7}\right) + 5 \]
Simplify:
\[ x = -\frac{34}{7} + 5 \]
Convert 5 to a fraction with a denominator of 7:
\[ 5 = \frac{35}{7} \]
So:
\[ x = -\frac{34}{7} + \frac{35}{7} \]
Combine the fractions:
\[ x = \frac{-34 + 35}{7} \]
\[ x = \frac{1}{7} \]
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####
Step 4: Write the solution as an ordered pair.
The solution to the system of equations is:
\[ (x, y) = \left( \frac{1}{7}, -\frac{17}{7} \right) \]
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Final Answer:
\[
\boxed{\left( \frac{1}{7}, -\frac{17}{7} \right)}
\]
Parent Tip: Review the logic above to help your child master the concept of gr 9 maths worksheets.
