Solve the given system of equations.

[tex]\[ \begin{array}{l}
2y = -x + 9 \\
3x - 6y = -15
\end{array} \][/tex]

The solution to the system is ( [tex]\(\square\)[/tex], [tex]\(\square\)[/tex] ).



Answer :

To solve the given system of equations, we can use the method of substitution. Here is a step-by-step solution:

Given system of equations:
[tex]\[ \begin{array}{l} 2y = -x + 9 \tag{1} \\ 3x - 6y = -15 \tag{2} \end{array} \][/tex]

Step 1: Solve equation (1) for [tex]\(y\)[/tex] in terms of [tex]\(x\)[/tex].

[tex]\[ 2y = -x + 9 \][/tex]
Divide both sides by 2 to isolate [tex]\(y\)[/tex]:

[tex]\[ y = \frac{-x + 9}{2} \][/tex]

Step 2: Substitute the expression for [tex]\(y\)[/tex] from Step 1 into equation (2).

[tex]\[ 3x - 6 \left(\frac{-x + 9}{2}\right) = -15 \][/tex]

Step 3: Simplify the substituted equation.

[tex]\[ 3x - 3(-x + 9) = -15 \][/tex]

Distribute the [tex]\(-3\)[/tex]:

[tex]\[ 3x + 3x - 27 = -15 \][/tex]

Combine like terms:

[tex]\[ 6x - 27 = -15 \][/tex]

Step 4: Solve for [tex]\(x\)[/tex].

[tex]\[ 6x = -15 + 27 \][/tex]

[tex]\[ 6x = 12 \][/tex]

[tex]\[ x = \frac{12}{6} \][/tex]

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

Step 5: Substitute [tex]\(x = 2\)[/tex] back into the expression for [tex]\(y\)[/tex] from Step 1.

[tex]\[ y = \frac{-x + 9}{2} \][/tex]

[tex]\[ y = \frac{-2 + 9}{2} \][/tex]

[tex]\[ y = \frac{7}{2} \][/tex]

[tex]\[ y = 3.5 \][/tex]

So, the solution to the system of equations is [tex]\((x, y) = (2, 3.5)\)[/tex].

Therefore, the solution to the system is [tex]\(\boxed{2}, \boxed{3.5}\)[/tex].

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