Question7. $\quad \begin{array}{l}2 x+y=2 \\ 3 x-2 y=-11\end{array}$

Studdy Solution

STEP 1

What is this asking? We've got two equations with two mystery numbers, $x$ and $y$ , and we need to find the values of both! Watch out! Be super careful with your positive and negative signs when rearranging and combining equations.
One little slip-up, and your answers will be way off!

STEP 2

1. Set up for elimination
2. Eliminate $y$
3. Solve for $x$
4. Substitute and solve for $y$

STEP 3

We're going to use elimination to solve this system of equations.
That means we want to tweak the equations so that when we add them together, either $x$ or $y$ disappears!
Let's aim to eliminate $y$ .

STEP 4

Look at the equations: $2x + y = 2$ $3x - 2y = -11$ Notice that the first equation has $y$ and the second has $-2y$ .
If we multiply the entire first equation by 2, we'll have $2y$ in the first equation, and when we add it to the $-2y$ in the second equation, the $y$ terms will add to zero and poof be gone!

STEP 5

So, let's multiply both sides of the first equation by 2: $2 \cdot (2x + y) = 2 \cdot 2$ $4x + 2y = 4$

STEP 6

Now, add the modified first equation to the second equation: $(4x + 2y) + (3x - 2y) = 4 + (-11)$ $4x + 2y + 3x - 2y = -7$ $7x = -7$ See how the $y$ terms vanished? Magic!

STEP 7

We're left with $7x = -7$ .
To solve for $x$ , we need to divide both sides of the equation by 7: $\frac{7x}{7} = \frac{-7}{7}$ $x = -1$ Boom! We found $x$ !

STEP 8

Now, let's plug our shiny new value for $x$ (which is $-1$ ) back into one of the original equations.
Let's use the first original equation, $2x + y = 2$ : $2(-1) + y = 2$ $-2 + y = 2$

STEP 9

To isolate $y$ , we need to add 2 to both sides of the equation: $-2 + y + 2 = 2 + 2$ $y = 4$ And there you have it!
We've found both $x$ and $y$ !

STEP 10

$x = -1$ and $y = 4$ .

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