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Solve the system of equations:

[tex]\[
\begin{cases}
x + 2y = 12 \\
y = -3x + 11
\end{cases}
\][/tex]


Sagot :

Certainly! Let's solve the given system of linear equations step-by-step:

[tex]\[ \begin{cases} x + 2y = 12 \\ y = -3x + 11 \end{cases} \][/tex]

Step 1: Express one variable in terms of the other.

From the second equation, we already have [tex]\( y \)[/tex] expressed in terms of [tex]\( x \)[/tex]:

[tex]\[ y = -3x + 11 \][/tex]

Step 2: Substitute this expression into the first equation.

Substitute [tex]\( y = -3x + 11 \)[/tex] into the first equation:

[tex]\[ x + 2(-3x + 11) = 12 \][/tex]

Step 3: Simplify and solve for [tex]\( x \)[/tex].

Distribute the 2 through the equation:

[tex]\[ x + 2(-3x) + 2(11) = 12 \][/tex]

[tex]\[ x - 6x + 22 = 12 \][/tex]

Combine like terms:

[tex]\[ -5x + 22 = 12 \][/tex]

Isolate [tex]\( x \)[/tex] by subtracting 22 from both sides:

[tex]\[ -5x = 12 - 22 \][/tex]

[tex]\[ -5x = -10 \][/tex]

Divide both sides by -5:

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

Step 4: Substitute [tex]\( x \)[/tex] back into the expression for [tex]\( y \)[/tex].

Now that we have [tex]\( x = 2 \)[/tex], substitute it into the expression for [tex]\( y \)[/tex]:

[tex]\[ y = -3(2) + 11 \][/tex]

[tex]\[ y = -6 + 11 \][/tex]

[tex]\[ y = 5 \][/tex]

Step 5: State the solution.

The solution to the system of equations is:

[tex]\[ (x, y) = (2, 5) \][/tex]

So, the values of [tex]\( x \)[/tex] and [tex]\( y \)[/tex] that satisfy both equations are [tex]\( x = 2 \)[/tex] and [tex]\( y = 5 \)[/tex].
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