Solve the system of equations using elimination: -3x + 2y = 10 and 5x - 5y = -10.
Understand the Problem
The question is asking us to solve a system of two equations using the elimination method. The equations provided are -3x + 2y = 10 and 5x - 5y = -10. We will need to eliminate one of the variables to find the values of x and y.
Answer
The solution is \( x = -6 \) and \( y = -4 \).
Answer for screen readers
The solution to the system of equations is ( x = -6 ) and ( y = -4 ).
Steps to Solve
- Align the equations for elimination
We have the following two equations:
$$ -3x + 2y = 10 \tag{1} $$
$$ 5x - 5y = -10 \tag{2} $$
- Manipulate one equation to align coefficients
To eliminate $y$, we'll modify equation (1). We can multiply equation (1) by 5 to make the coefficient of $y$ the same (10):
$$ 5(-3x + 2y) = 5(10) \Rightarrow -15x + 10y = 50 \tag{3} $$
- Manipulate the second equation
Now we can multiply equation (2) by 2:
$$ 2(5x - 5y) = 2(-10) \Rightarrow 10x - 10y = -20 \tag{4} $$
- Add equations to eliminate $y$
Now we'll add equations (3) and (4):
$$ (-15x + 10y) + (10x - 10y) = 50 - 20 $$
This simplifies to:
$$ -5x = 30 $$
- Solve for $x$
Now, we can solve for $x$:
$$ x = \frac{30}{-5} = -6 $$
- Substitute $x$ back into one of the original equations
We can use equation (1) to find $y$:
$$ -3(-6) + 2y = 10 \Rightarrow 18 + 2y = 10 $$
- Solve for $y$
Rearranging gives:
$$ 2y = 10 - 18 \Rightarrow 2y = -8 \Rightarrow y = \frac{-8}{2} = -4 $$
The solution to the system of equations is ( x = -6 ) and ( y = -4 ).
More Information
These values of ( x ) and ( y ) satisfy both original equations. The elimination method is useful for systematically removing one variable to solve for the other.
Tips
- Failing to multiply one or both equations correctly, leading to incorrect coefficients.
- Not aligning the equations properly before addition or subtraction.
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