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Find the solution to the following system of equations using substitution:

[tex]\[
\begin{cases}
x - 3y = 2 \\
2x + 5y = 15
\end{cases}
\][/tex]



Answer :

GinnyAnswer
Sure, let's solve the given system of equations step-by-step using substitution:

[tex]\[ \begin{aligned} (1) & \quad x - 3y = 2 \\ (2) & \quad 2x + 5y = 15 \end{aligned} \][/tex]

### Step 1: Solve one of the equations for one variable
Let's solve equation (1) for [tex]\( x \)[/tex]:

[tex]\[ x - 3y = 2 \quad \Rightarrow \quad x = 2 + 3y \][/tex]

### Step 2: Substitute this expression into the other equation
Now we'll substitute [tex]\( x = 2 + 3y \)[/tex] into equation (2):

[tex]\[ 2(2 + 3y) + 5y = 15 \][/tex]

### Step 3: Simplify and solve for [tex]\( y \)[/tex]
First, distribute the 2 on the left side:

[tex]\[ 4 + 6y + 5y = 15 \][/tex]

Combine like terms:

[tex]\[ 4 + 11y = 15 \][/tex]

Subtract 4 from both sides to isolate the term with [tex]\( y \)[/tex]:

[tex]\[ 11y = 11 \][/tex]

Divide both sides by 11:

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

### Step 4: Substitute [tex]\( y \)[/tex] back into the expression for [tex]\( x \)[/tex]
We have found [tex]\( y = 1 \)[/tex]. Substitute this value back into the expression [tex]\( x = 2 + 3y \)[/tex]:

[tex]\[ x = 2 + 3(1) \quad \Rightarrow \quad x = 2 + 3 \quad \Rightarrow \quad x = 5 \][/tex]

### Step 5: Write the solution
The solution to the system of equations is:

[tex]\[ \boxed{(x, y) = (5, 1)} \][/tex]

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