The graph of the equation px+qy=12px+qy=12 is a line in the xyxy-plane, where pp and qq are constants. If the line passes through the points (−1,1)(−1,1) and (2,3)(2,3), what is the value of qq?

To find the value of \( q \), we need to use the fact that the line passes through the points \((-1, 1)\) and \( (2, 3) \). The equation of the line is given as:

\[
px + qy = 12
\]

### Step 1: Substitute the points into the equation

First, substitute the point \((-1, 1)\) into the equation:

\[
p(-1) + q(1) = 12 \implies -p + q = 12 \quad \text{(Equation 1)}
\]

Next, substitute the point \( (2, 3) \) into the equation:

\[
p(2) + q(3) = 12 \implies 2p + 3q = 12 \quad \text{(Equation 2)}
\]

### Step 2: Solve the system of equations

We have the system of linear equations:

\[
\begin{aligned}
- p + q &= 12 \quad \text{(Equation 1)} \\
2p + 3q &= 12 \quad \text{(Equation 2)}
\end{aligned}
\]

#### Solve Equation 1 for \( p \):
\[
p = q - 12 \quad \text{(Equation 3)}
\]

#### Substitute Equation 3 into Equation 2:
\[
2(q - 12) + 3q = 12
\]

Expand and simplify:
\[
2q - 24 + 3q = 12
\]
\[
5q - 24 = 12
\]

#### Solve for \( q \):
\[
5q = 36
\]
\[
q = \frac{36}{5} = 7.2
\]

### Final Answer
The value of \( q \) is \( 7.2 \).

Would you like further explanation, or do you have any questions about the process?

Here are 5 related questions:
1. How do you solve a system of linear equations with two variables?
2. Can you find the value of \( p \) using the same method?
3. What is the slope of the line given by the equation \( px + qy = 12 \)?
4. How would the solution change if the line passed through different points?
5. Can you express the equation in slope-intercept form \( y = mx + b \)?

**Tip:** When solving for one variable, substitution is a powerful method, especially if one equation is easy to rearrange.

Learn how to solve a system of linear equations using points (-1, 1) and (2, 3). This problem involves finding constants p and q in the equation px + qy = 12. Suitable for students in Grades 7-9.

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