tofi69
Answered

Solve the system of equations:

[tex]\[
\begin{cases}
2x + y = -5 \\
2x - 5y = 13
\end{cases}
\][/tex]



Answer :

To solve the system of linear equations given by:

[tex]\[ \begin{cases} 2x + y = -5 \\ 2x - 5y = 13 \end{cases} \][/tex]

we will use the method of elimination to find the values of [tex]\(x\)[/tex] and [tex]\(y\)[/tex].

1. Step 1: Align the two equations:
[tex]\[ \begin{cases} 2x + y = -5 \quad \text{(Equation 1)} \\ 2x - 5y = 13 \quad \text{(Equation 2)} \end{cases} \][/tex]

2. Step 2: Eliminate one variable:
Since both equations have the term [tex]\(2x\)[/tex], we can eliminate [tex]\(x\)[/tex] by subtracting Equation 1 from Equation 2.

Subtract Equation 1 from Equation 2:
[tex]\[ (2x - 5y) - (2x + y) = 13 - (-5) \][/tex]

Simplifying the left-hand side and the right-hand side:
[tex]\[ 2x - 5y - 2x - y = 13 + 5 \][/tex]

[tex]\[ -6y = 18 \][/tex]

3. Step 3: Solve for [tex]\(y\)[/tex]:
Divide both sides by [tex]\(-6\)[/tex]:
[tex]\[ y = \frac{18}{-6} \][/tex]

[tex]\[ y = -3 \][/tex]

4. Step 4: Substitute [tex]\(y\)[/tex] back into one of the original equations to find [tex]\(x\)[/tex]:
We can use Equation 1:
[tex]\[ 2x + y = -5 \][/tex]

Substitute [tex]\(y = -3\)[/tex]:
[tex]\[ 2x + (-3) = -5 \][/tex]

Simplify:
[tex]\[ 2x - 3 = -5 \][/tex]

Add 3 to both sides:
[tex]\[ 2x = -5 + 3 \][/tex]

[tex]\[ 2x = -2 \][/tex]

5. Step 5: Solve for [tex]\(x\)[/tex]:
Divide both sides by 2:
[tex]\[ x = \frac{-2}{2} \][/tex]

[tex]\[ x = -1 \][/tex]

Therefore, the solution to the system of equations is:

[tex]\[ x = -1 \quad \text{and} \quad y = -3 \][/tex]

So, the ordered pair [tex]\((x, y)\)[/tex] is [tex]\((-1, -3)\)[/tex].