Solve the system of equations using elimination: -2x - y = 13 and 7x + 3y = -42.
Understand the Problem
The question is asking to solve a system of equations using the elimination method. It provides two equations: -2x - y = 13 and 7x + 3y = -42, and requires finding the values of x and y that satisfy both equations simultaneously.
Answer
The solution is \( (x, y) = (-3, -7) \).
Answer for screen readers
The solution to the system of equations is ( (x, y) = (-3, -7) ).
Steps to Solve
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Set Up the Equations The given equations are: $$ -2x - y = 13 $$ $$ 7x + 3y = -42 $$
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Multiply to Align Coefficients To eliminate $y$, we can multiply the first equation by 3: $$ 3(-2x - y) = 3(13) $$ This results in: $$ -6x - 3y = 39 $$
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Rewrite the System of Equations Now we have the new system: $$ -6x - 3y = 39 $$ $$ 7x + 3y = -42 $$
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Add the Equations to Eliminate $y$ Now, add the two equations: $$ (-6x - 3y) + (7x + 3y) = 39 + (-42) $$ This simplifies to: $$ x = -3 $$
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Substitute $x$ Back to Find $y$ Substitute $x = -3$ into one of the original equations. Using the first equation: $$ -2(-3) - y = 13 $$ This simplifies to: $$ 6 - y = 13 $$
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Solve for $y$ Rearranging gives: $$ -y = 13 - 6 $$ So, $$ -y = 7 $$ Then, multiplying by -1 results in: $$ y = -7 $$
The solution to the system of equations is ( (x, y) = (-3, -7) ).
More Information
The elimination method is useful for solving systems of linear equations by reducing them to simpler forms. This method effectively allows us to isolate one variable, making it easier to solve for the other.
Tips
- Forgetting to multiply correctly when aligning coefficients can lead to incorrect equations.
- Making mistakes with signs during substitution or addition of equations can lead to errors in the solution.
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