https://r2.erweima.ai/mathbot/image/17260152813852vdiafzg.jpg

The system of equations provided in the image is:

\[
\begin{aligned}
5x + 2y &= 1 \\
-3x + 3y &= 5
\end{aligned}
\]

We are tasked with solving this system using the **substitution method**. Let's go step by step:

### Step 1: Solve one equation for one variable
We will solve the first equation for \(x\):

\[
5x + 2y = 1
\]
\[
5x = 1 - 2y
\]
\[
x = \frac{1 - 2y}{5}
\]

### Step 2: Substitute the expression for \(x\) into the second equation
Now, substitute \(x = \frac{1 - 2y}{5}\) into the second equation \(-3x + 3y = 5\):

\[
-3\left(\frac{1 - 2y}{5}\right) + 3y = 5
\]

### Step 3: Solve for \(y\)
Simplify the equation:

\[
-\frac{3(1 - 2y)}{5} + 3y = 5
\]
\[
-\frac{3 - 6y}{5} + 3y = 5
\]
Multiply the entire equation by 5 to eliminate the denominator:

\[
-(3 - 6y) + 15y = 25
\]
Distribute the negative sign:

\[
-3 + 6y + 15y = 25
\]
Combine like terms:

\[
21y - 3 = 25
\]
Add 3 to both sides:

\[
21y = 28
\]
Divide by 21:

\[
y = \frac{28}{21} = \frac{4}{3}
\]

### Step 4: Solve for \(x\)
Now that we have \(y = \frac{4}{3}\), substitute this value back into the equation for \(x\):

\[
x = \frac{1 - 2\left(\frac{4}{3}\right)}{5}
\]
\[
x = \frac{1 - \frac{8}{3}}{5}
\]
\[
x = \frac{\frac{3}{3} - \frac{8}{3}}{5} = \frac{-\frac{5}{3}}{5}
\]
\[
x = -\frac{5}{15} = -\frac{1}{3}
\]

### Final Solution:
The solution to the system is:
\[
x = -\frac{1}{3}, \quad y = \frac{4}{3}
\]

Would you like more details on this solution or have any questions?

Here are 5 related questions you might want to consider:
1. How would the solution process change if we used the elimination method instead of substitution?
2. Can you solve a system with no solutions using substitution? How would you recognize it?
3. How does solving a system graphically compare to using substitution?
4. What are the advantages of the substitution method over other methods like elimination or matrix solutions?
5. What happens if you switch the order of solving for \(x\) or \(y\) first? Would the solution change?

**Tip**: When choosing a variable to solve for in substitution, try to pick one that minimizes fractions to simplify the calculations.

Solve by substitution: 5x + 2y = 1, -3x + 3y = 5

Learn how to solve the system of equations 5x + 2y = 1 and -3x + 3y = 5 using the substitution method. This step-by-step solution breaks down solving for one variable and substituting it into the second equation. Ideal for students in Grades 8-10 learning algebra.

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