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Solve the system of equations:

[tex]\[
\begin{cases}
x + 3y = 20 \\
x + y = 12
\end{cases}
\][/tex]

Sagot :

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

[tex]\[ \left\{\begin{array}{r} x + 3y = 20 \\ x + y = 12 \end{array}\right. \][/tex]

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

First, we can solve the second equation for [tex]\( x \)[/tex] in terms of [tex]\( y \)[/tex]:

[tex]\[ x + y = 12 \implies x = 12 - y \][/tex]

### Step 2: Substitute the expression into the first equation

Next, we'll substitute [tex]\( x = 12 - y \)[/tex] into the first equation:

[tex]\[ x + 3y = 20 \][/tex]

[tex]\[ (12 - y) + 3y = 20 \][/tex]

### Step 3: Simplify the resulting equation

Combine like terms:

[tex]\[ 12 - y + 3y = 20 \][/tex]

[tex]\[ 12 + 2y = 20 \][/tex]

### Step 4: Solve for [tex]\( y \)[/tex]

Isolate [tex]\( y \)[/tex] by subtracting 12 from both sides:

[tex]\[ 2y = 20 - 12 \][/tex]

[tex]\[ 2y = 8 \][/tex]

[tex]\[ y = 4 \][/tex]

### Step 5: Substitute [tex]\( y = 4 \)[/tex] back into the expression for [tex]\( x \)[/tex]

Now substitute [tex]\( y = 4 \)[/tex] back into the equation [tex]\( x = 12 - y \)[/tex]:

[tex]\[ x = 12 - 4 \][/tex]

[tex]\[ x = 8 \][/tex]

### Conclusion

The solution to the system of equations is:

[tex]\[ x = 8, \quad y = 4 \][/tex]

Therefore, the values of [tex]\( x \)[/tex] and [tex]\( y \)[/tex] that satisfy both equations are [tex]\( x = 8 \)[/tex] and [tex]\( y = 4 \)[/tex].
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