Question $\begin{array}{l}3 x-y=-7 \\ 4 x+2 y=-6\end{array}$

Studdy Solution

STEP 1

1. We are given a system of two linear equations with two variables, $x$ and $y$ .
2. The goal is to find the values of $x$ and $y$ that satisfy both equations simultaneously.
3. We will use the method of substitution or elimination to solve the system.

STEP 2

1. Choose a method to solve the system of equations.
2. Solve for one variable in terms of the other.
3. Substitute the expression into the other equation.
4. Solve for the remaining variable.
5. Substitute back to find the other variable.
6. Verify the solution by substituting both values into the original equations.

STEP 3

We will use the method of elimination to solve the system of equations.

STEP 4

First, let's eliminate $y$ by making the coefficients of $y$ in both equations equal. Multiply the first equation by 2:
$2(3x - y) = 2(-7)$
This simplifies to:
$6x - 2y = -14$

STEP 5

Now, we have the system:
$\begin{array}{l} 6x - 2y = -14 \\ 4x + 2y = -6 \end{array}$
Add the two equations together to eliminate $y$ :
$(6x - 2y) + (4x + 2y) = -14 + (-6)$
This simplifies to:
$10x = -20$

STEP 6

Solve for $x$ by dividing both sides by 10:
$x = \frac{-20}{10} = -2$

STEP 7

Substitute $x = -2$ back into the first equation to solve for $y$ :
$3(-2) - y = -7$
Simplify:
$-6 - y = -7$
Add 6 to both sides:
$-y = -1$
Multiply both sides by -1:
$y = 1$

STEP 8

Verify the solution by substituting $x = -2$ and $y = 1$ into the second original equation:
$4(-2) + 2(1) = -6$
Simplify:
$-8 + 2 = -6$
Both sides are equal, confirming our solution is correct.
The solution to the system is:
$(x, y) = (-2, 1)$

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