Solve the following system of equations using elimination: x + 4y = 20 3x + 5y = 11
Understand the Problem
The question asks to solve a system of two linear equations with two variables (x and y) using the elimination method. The goal is to find the values of x and y that satisfy both equations simultaneously.
Answer
$(-8, 7)$
Answer for screen readers
$(-8, 7)$
Steps to Solve
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Multiply the first equation by -3 Multiply each term in the equation $x + 4y = 20$ by $-3$ to get $-3x - 12y = -60$.
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Write the modified system of equations The new system of equations is: $$ -3x - 12y = -60 $$ $$ 3x + 5y = 11 $$
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Add the equations together Add the two equations together to eliminate the $x$ variable: $$ (-3x - 12y) + (3x + 5y) = -60 + 11 $$ $$ -7y = -49 $$
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Solve for y Divide both sides of the equation by $-7$ to find the value of $y$: $$ y = \frac{-49}{-7} = 7 $$
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Substitute the value of y into the first original equation Substitute $y = 7$ into the equation $x + 4y = 20$: $$ x + 4(7) = 20 $$ $$ x + 28 = 20 $$
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Solve for x Subtract 28 from both sides of the equation to find the value of $x$: $$ x = 20 - 28 = -8 $$
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Write the solution as an ordered pair The solution to the system of equations is $x = -8$ and $y = 7$. Write the solution as an ordered pair, $(x, y) = (-8, 7)$.
$(-8, 7)$
More Information
The elimination method is a useful technique for solving systems of linear equations. It involves manipulating the equations so that when they are added together, one of the variables cancels out, making it easier to solve for the remaining variable.
Tips
A common mistake is not distributing the multiplication correctly in step 1. For example, forgetting to multiply the constant term. Another common mistake is making an arithmetic error when adding the equations or solving for $x$ or $y$. Always double-check your work to avoid these mistakes.
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