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Solve the system of equations:
[tex]\[
\begin{aligned}
5x - 3y &= 22 \\
x - y &= 4
\end{aligned}
\][/tex]

Sagot :

GinnyAnswer
Let's solve the given system of linear equations using the method of substitution or elimination. Here are the equations:

[tex]\[ \begin{aligned} 5x - 3y &= 22 \quad \quad (1) \\ x - y &= 4 \quad \quad \quad (2) \end{aligned} \][/tex]

### Step-by-Step Solution

#### Step 1: Solving one of the equations for one variable

Let's solve Equation (2) for [tex]\( x \)[/tex]:

[tex]\[ x - y = 4 \][/tex]
[tex]\[ x = y + 4 \quad \quad \quad (3) \][/tex]

#### Step 2: Substituting into the other equation

Now substitute Equation (3) into Equation (1):

[tex]\[ 5(y + 4) - 3y = 22 \][/tex]

#### Step 3: Simplifying the substituted equation

Expand and simplify:

[tex]\[ 5y + 20 - 3y = 22 \][/tex]
[tex]\[ 2y + 20 = 22 \][/tex]
[tex]\[ 2y = 22 - 20 \][/tex]
[tex]\[ 2y = 2 \][/tex]

Divide both sides by 2 to solve for [tex]\( y \)[/tex]:

[tex]\[ y = 1 \][/tex]

#### Step 4: Finding the value of [tex]\( x \)[/tex]

Now that we have [tex]\( y = 1 \)[/tex], substitute this value back into Equation (3) to find [tex]\( x \)[/tex]:

[tex]\[ x = 1 + 4 \][/tex]
[tex]\[ x = 5 \][/tex]

### Solution

The solution to the system of equations is:

[tex]\[ x = 5 \][/tex]
[tex]\[ y = 1 \][/tex]

Hence, the values that satisfy both equations are [tex]\( x = 5 \)[/tex] and [tex]\( y = 1 \)[/tex].
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