Question $\begin{array}{l}x-5 y=15 \\ -x+3 y=-11\end{array}$

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 elimination to solve the system.

STEP 2

1. Write down the system of equations.
2. Add the equations to eliminate one variable.
3. Solve for the remaining variable.
4. Substitute the value back into one of the original equations to find the other variable.
5. Verify the solution by substituting both values back into the original equations.

STEP 3

Write down the system of equations:
\begin{align}
1. & \quad x - 5y = 15 \\
2. & \quad -x + 3y = -11 \end{align}

STEP 4

Add the equations to eliminate $x$ :
$(x - 5y) + (-x + 3y) = 15 + (-11)$
This simplifies to:
$-2y = 4$

STEP 5

Solve for $y$ :
$-2y = 4$
Divide both sides by $-2$ :
$y = -2$

STEP 6

Substitute $y = -2$ back into one of the original equations to find $x$ . Using the first equation:
$x - 5(-2) = 15$
Simplify and solve for $x$ :
$x + 10 = 15$
Subtract 10 from both sides:
$x = 5$

STEP 7

Verify the solution by substituting $x = 5$ and $y = -2$ back into the original equations:
For the first equation:
$5 - 5(-2) = 15 \quad \Rightarrow \quad 5 + 10 = 15 \quad \Rightarrow \quad 15 = 15 \quad \text{True}$
For the second equation:
$-5 + 3(-2) = -11 \quad \Rightarrow \quad -5 - 6 = -11 \quad \Rightarrow \quad -11 = -11 \quad \text{True}$
Both equations are satisfied.
The solution to the system is $x = 5$ and $y = -2$ .

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