Solve the system of equations using elimination: 7x - 2y = 11 and x - 3y = 7.

Understand the Problem

The question is asking to solve a system of equations using the elimination method. The equations provided are 7x - 2y = 11 and x - 3y = 7. We will combine these equations in a way to eliminate one of the variables and solve for the other.

Answer

The solution is $x = 1$ and $y = -2$.

Steps to Solve

Prepare the equations for elimination

The given equations are: $$ 7x - 2y = 11 \quad \text{(1)} $$ $$ x - 3y = 7 \quad \text{(2)} $$

We can multiply equation (2) by 7 so that the coefficient of $x$ in both equations is the same:

$$ 7(x - 3y) = 7 \cdot 7 $$ This results in: $$ 7x - 21y = 49 \quad \text{(3)} $$

Set up the new system of equations

Now we will have a new system: $$ 7x - 2y = 11 \quad \text{(1)} $$ $$ 7x - 21y = 49 \quad \text{(3)} $$

Eliminate $x$ by subtracting the equations

To eliminate $x$, we will subtract equation (1) from equation (3):

$$ (7x - 21y) - (7x - 2y) = 49 - 11 $$ This simplifies to: $$ -21y + 2y = 38 $$ Which further simplifies to: $$ -19y = 38 $$

Solve for $y$

Now we will isolate $y$:

$$ y = \frac{38}{-19} = -2 $$

Substitute $y$ back into one of the original equations

Let's substitute $y = -2$ back into equation (2):

$$ x - 3(-2) = 7 $$ This simplifies to: $$ x + 6 = 7 $$ Now we solve for $x$: $$ x = 7 - 6 = 1 $$

The solution to the system of equations is $x = 1$ and $y = -2$.

More Information

This system of equations has a single unique solution, which means that the lines represented by the equations intersect at the point (1, -2) on the coordinate plane.

Tips

Forgetting to multiply both sides of an equation when scaling them to align coefficients.
Failing to correctly change signs when performing subtraction.

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