Solve the following system of equations using elimination: -x - y = 1 -2x - y = -6
Understand the Problem
The problem requires solving a system of two linear equations using the elimination method. The goal is to find the values of 'x' and 'y' that satisfy both equations simultaneously.
Answer
$(7, -8)$
Answer for screen readers
$(7, -8)$
Steps to Solve
- Multiply the first equation by -1
Multiply the first equation by -1 to change the sign of each term, making it easier to eliminate $y$.
$$ (-1) * (-x - y) = (-1) * (1) $$ $$ x + y = -1 $$
- Write the new system of equations
Now our system of equations is: $$ x + y = -1 $$ $$ -2x - y = -6 $$
- Eliminate y by adding the two equations
Add the two equations together. The $y$ terms will cancel out ($y + (-y) = 0$). $$ (x + y) + (-2x - y) = -1 + (-6) $$ $$ x - 2x + y - y = -7 $$ $$ -x = -7 $$
- Solve for x
Solve for $x$ by dividing both sides of the equation by -1. $$ -x = -7 $$ $$ x = 7 $$
- Substitute the value of x into one of the original equations to solve for y
Substitute $x = 7$ into the first original equation: $$ -x - y = 1 $$ $$ -7 - y = 1 $$
- Isolate y
Add 7 to both sides: $$ -y = 1 + 7 $$ $$ -y = 8 $$ $$ y = -8 $$
- State the solution
The solution to the system of equations is $x = 7$ and $y = -8$.
$(7, -8)$
More Information
The elimination method is a technique used to solve systems of linear equations by eliminating one of the variables. In this case, we eliminated $y$ to solve for $x$, then substituted the value of $x$ back into one of the original equations to solve for $y$.
Tips
A common mistake is making errors when distributing the negative sign while multiplying equations, especially in step 1. For example, forgetting to multiply the constant term on the right side of the equation. Another common mistake is making arithmetic errors when adding or subtracting the equations in the elimination step.
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