2021 Maths Standard 2 HSC Q34 How to model, graph, solve simultaneous linear relationships/equations

Source: © NSW Education Standards Authority

Disclaimer: This sample solution is intended as a guide only and does not necessarily represent the best way to answer the question, nor is there any guarantee of accuracy.
~
Modeling a linear relationship using algebra involves representing the relationship between two variables with a linear equation. A linear equation has the general form:

y = mx + c

where:
- y is the dependent variable (the one you're trying to predict or explain),
- x is the independent variable (the one you're using to make predictions),
- m is the gradient (slope) of the line (the rate of change of y with respect to x),
- b is the y-intercept (the value of \( y \) when \( x \) is zero).

Here's a step-by-step guide to modeling a linear relationship using algebra:

1. **Identify the Variables:**
- Determine which variable is the independent variable (x) and which is the dependent variable (y).

2. **Collect Data:**
- Gather data points that represent the relationship between x and y. Each data point should have corresponding values for both variables.

3. **Calculate the gradient (m):**
- Use the formula for gradient: m = (change in y) \ (change in x) [OR m = rise over run]
- Choose two points from your data and substitute in their coordinates to find the slope.

4. **Calculate the y-Intercept (c):**
- Choose one of the points from your data and use it to solve for c in the equation y = mx + c.

5. **Write the Linear Equation:**
- Once you have m and c, write the linear equation in the form y = mx + c (gradient-intercept form).

6. **Interpret the Equation:**
- Understand the meaning of the gradient (m) and the y-intercept (c) in the context of your problem. The gradient represents the rate of change, and the y-intercept is the value of y when x is zero.

7. **Use the Model for Predictions:**
- With the linear equation, you can now make predictions for y based on given values of x.

For example, if your data suggests a linear relationship between the number of hours studied (x) and the score on a test (y), your equation might be something like y = 2x + 70. This would imply that for every additional hour studied (x), the expected test score (y) increases by 2 points, and the expected score when no hours are studied is 70.

Keep in mind that this is a basic guide, and there are more advanced techniques for modeling relationships in more complex situations.

Graphing a linear relationship in the form y = mx + c involves plotting points on a coordinate plane that satisfy the equation. Here are the steps to graph a linear equation:

1. **Understand the Equation:**
- y = mx + c represents a line in the coordinate plane, where:
- m is the gradient (the rate of change of y with respect to x),
- c is the y-intercept (the value of y when x = 0).

2. **Identify the Gradient and y-Intercept:**
- m is the coefficient of x, and c is the constant term.

3. **Plot the y-Intercept:**
- The y-intercept is the point where the line crosses the y-axis. Plot the point (0, c).

4. **Use the Slope to Plot Additional Points:**
- The gradient m represents the "rise" over "run." This means that for every increase of 1 in x, y increases by m. Move up or down by the gradient and to the right by 1 to find another point on the line. Repeat this process to plot more points.

5. **Draw the Line:**
- Connect the plotted points with a straight line. The line represents the linear relationship described by the equation.

### Example:

Consider the equation y = 2x + 3.

**Step 1:** Understand the equation. It's in the form y = mx + c, where m = 2 and b = 3.

**Step 2:** Identify the gradient (m) and y-intercept (c).

**Step 3:** Plot the y-intercept, which is the point (0, 3).

**Step 4:** Use the gradient to plot additional points. Since m = 2, move up 2 and to the right 1 from the y-intercept to find another point, and repeat this process.

**Step 5:** Draw the line through the plotted points.

The resulting graph is a straight line that represents the linear relationship described by the equation y = 2x + 3.