Solve the system of equations using elimination: 2x + y = -10 and -3x - 5y = -20. Find the values of x and y where they intercept.

Understand the Problem

The question is asking to solve a system of linear equations using the elimination method. We are given two equations: 2x + y = -10 and -3x - 5y = -20, and we need to find the values of x and y where these two equations intersect.

Answer

The solution is $x = -10$, $y = 10$.

Steps to Solve

Multiply to Align Coefficients

To eliminate one variable, we will multiply the first equation by 5. This will help align the coefficients of $y$.

Starting with:

$$ 2x + y = -10 $$

Multiply by 5:

$$ 10x + 5y = -50 $$

Now we have:

$10x + 5y = -50$
$-3x - 5y = -20$

Add the Equations

Now add the two equations together to eliminate $y$:

$$ (10x + 5y) + (-3x - 5y) = -50 - 20 $$

This simplifies to:

$$ 7x = -70 $$

Solve for x

Next, we solve for $x$ by dividing both sides by 7:

$$ x = \frac{-70}{7} = -10 $$

Substitute x to Find y

Now that we have $x = -10$, we substitute this value back into the first equation to find $y$:

$$ 2(-10) + y = -10 $$

This simplifies to:

$$ -20 + y = -10 $$

So we can solve for $y$:

$$ y = -10 + 20 = 10 $$

State the Solution

The values of $x$ and $y$ that solve the system are $x = -10$ and $y = 10$.

The solution is $x = -10$ and $y = 10$.

More Information

In this problem, we solved a system of linear equations using the elimination method, which is an effective way to eliminate variables and find solutions. The result indicates the intersection point of the two lines represented by the equations.

Tips

Common mistakes in this problem include:

Not aligning the coefficients correctly when preparing to add the equations.
Making arithmetic errors when adding or subtracting numbers, particularly when negative signs are involved.
Forgetting to substitute back into one of the original equations to find the second variable.

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