solve system of equations: 3x-5y-7=0 and 6x+7y=8

Let's solve the system of equations:

\[
\begin{aligned}
1. & \quad 3x - 5y - 7 = 0 \\
2. & \quad 6x + 7y = 8
\end{aligned}
\]

### Step 1: Simplify the first equation

Start by rearranging the first equation to express it in the standard form:

\[
3x - 5y = 7
\]

### Step 2: Express one variable in terms of the other

Now let's express \(x\) from the first equation:

\[
3x = 5y + 7 \quad \Rightarrow \quad x = \frac{5y + 7}{3}
\]

### Step 3: Substitute \(x\) into the second equation

Substitute \(x = \frac{5y + 7}{3}\) into the second equation:

\[
6\left(\frac{5y + 7}{3}\right) + 7y = 8
\]

Simplify the equation:

\[
2(5y + 7) + 7y = 8
\]

\[
10y + 14 + 7y = 8
\]

### Step 4: Solve for \(y\)

Combine like terms:

\[
17y + 14 = 8
\]

Isolate \(y\):

\[
17y = 8 - 14
\]

\[
17y = -6
\]

\[
y = -\frac{6}{17}
\]

### Step 5: Solve for \(x\)

Now substitute \(y = -\frac{6}{17}\) back into the expression for \(x\):

\[
x = \frac{5\left(-\frac{6}{17}\right) + 7}{3}
\]

Simplify:

\[
x = \frac{-\frac{30}{17} + \frac{119}{17}}{3}
\]

\[
x = \frac{\frac{89}{17}}{3} = \frac{89}{51}
\]

### Final Answer:
\[
x = \frac{89}{51}, \quad y = -\frac{6}{17}
\]

Would you like more details or have any questions? Here are some related questions to consider:

1. What would happen if both equations were multiplied by a constant before solving?
2. How can we verify the solution graphically?
3. What is the geometric interpretation of this solution?
4. How does substitution compare to other methods like elimination for this system?
5. How does solving with fractions impact the accuracy of your answer?

**Tip:** When dealing with fractions, it's often helpful to simplify them at each step to avoid mistakes later in the calculation.

Learn how to solve the system of equations 3x - 5y - 7 = 0 and 6x + 7y = 8 step-by-step with a detailed solution. This problem involves linear equations, substitution method, and operations with fractions. Suitable for students in Grades 9-12.

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