Solve the system of equations.

3x−5y=4

−2x+6y=18


Show your work. Write your answer as (x, y).

We are given the system of equations:

\[
3x - 5y = 4 \tag{1}
\]
\[
-2x + 6y = 18 \tag{2}
\]

### Step 1: Eliminate one variable

To eliminate one of the variables, we will multiply both equations by suitable values so that the coefficients of either \(x\) or \(y\) will cancel out when added.

Let’s eliminate \(x\). To do this, we can multiply equation (1) by 2 and equation (2) by 3, so that the coefficients of \(x\) will be opposites:

\[
2(3x - 5y) = 2(4) \implies 6x - 10y = 8 \tag{3}
\]
\[
3(-2x + 6y) = 3(18) \implies -6x + 18y = 54 \tag{4}
\]

### Step 2: Add the two equations

Now, add equations (3) and (4) to eliminate \(x\):

\[
(6x - 10y) + (-6x + 18y) = 8 + 54
\]
\[
0x + 8y = 62
\]
\[
8y = 62
\]

### Step 3: Solve for \(y\)

Now, solve for \(y\):

\[
y = \frac{62}{8} = \frac{31}{4}
\]

### Step 4: Substitute \(y = \frac{31}{4}\) into one of the original equations

We’ll substitute \(y = \frac{31}{4}\) into equation (1) to solve for \(x\):

\[
3x - 5\left(\frac{31}{4}\right) = 4
\]
\[
3x - \frac{155}{4} = 4
\]
Multiply through by 4 to eliminate the fraction:

\[
4(3x) - 155 = 16
\]
\[
12x = 16 + 155
\]
\[
12x = 171
\]
\[
x = \frac{171}{12} = \frac{57}{4}
\]

### Final Answer:

The solution to the system is:
\[
(x, y) = \left( \frac{57}{4}, \frac{31}{4} \right)
\]

Would you like further details or have any questions?

Here are 5 related questions to practice:

1. Solve the system: \( 4x + y = 7 \) and \( -3x + 2y = 4 \).
2. Solve the system using substitution: \( x + 2y = 5 \) and \( 3x - y = 2 \).
3. What is the graphical interpretation of solving a system of two linear equations?
4. Solve for \( x \) and \( y \) using elimination: \( 7x - 2y = 9 \) and \( -x + 4y = 3 \).
5. How would you check if a solution to a system of equations is correct?

**Tip**: When using the elimination method, always aim to eliminate the variable that gives the simplest arithmetic steps first.

Solve the system of equations.
3x−5y=4
−2x+6y=18

Show your work. Write your answer as (x, y).

Learn how to solve the system of equations 3x−5y=4 and −2x+6y=18 using the elimination method. This detailed solution involves algebraic manipulation, solving for variables, and substituting back to find the solution (x, y). Ideal for students in Grades 8-10.

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