Solve the system of equations using elimination: 3x + 3y = -9 and x + 3y = -19.
Understand the Problem
The question is asking to solve a system of equations using the elimination method. The system given is 3x + 3y = -9 and x + 3y = -19. We need to find the values of x and y by eliminating one of the variables.
Answer
The solution to the system is $(5, -8)$.
Answer for screen readers
The solution to the system of equations is:
$$ (x, y) = (5, -8) $$
Steps to Solve
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Write down the equations
We have the two equations:
$$ 3x + 3y = -9 $$
$$ x + 3y = -19 $$ -
Align the equations
The second equation can be rewritten to eliminate $y$ by using a method of elimination. Keeping both equations: $$ 3x + 3y = -9 $$
$$ x + 3y = -19 $$ -
Eliminate $y$
Subtract the second equation from the first. This allows us to eliminate $y$.
$$ (3x + 3y) - (x + 3y) = -9 - (-19) $$
Which simplifies to:
$$ 3x - x + 3y - 3y = -9 + 19 $$
$$ 2x = 10 $$ -
Solve for $x$
Now, divide both sides by 2:
$$ x = \frac{10}{2} = 5 $$ -
Substitute $x$ back into one of the original equations
Using the second equation:
$$ x + 3y = -19 $$
Substituting $x = 5$:
$$ 5 + 3y = -19 $$ -
Solve for $y$
Subtract 5 from both sides:
$$ 3y = -19 - 5 $$
$$ 3y = -24 $$
Then divide by 3:
$$ y = \frac{-24}{3} = -8 $$
The solution to the system of equations is:
$$ (x, y) = (5, -8) $$
More Information
In this system of equations, we used the elimination method to remove one variable and easily solve for the other. Elimination is a powerful technique that often simplifies finding solutions for two-variable systems.
Tips
- Forgetting to correctly align equations before subtraction.
- Miscalculating during subtraction or division steps.
- Using incorrect signs, especially when handling negatives.
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