Question

graphing linear equations

Answer 1

Graphing linear equations involves plotting points on a coordinate plane based on the equation of a line. The standard form of a linear equation is \(y = mx + b\), where:

- \(m\) is the slope of the line,
- \(b\) is the y-intercept (the point where the line crosses the y-axis).

Here’s a step-by-step guide to graphing a linear equation:

### Step 1: Identify the equation

Ensure your equation is in slope-intercept form (if not, rearrange it). For example, the equation \(2x + 3y = 6\) can be rearranged to \(y = -\frac{2}{3}x + 2\).

### Step 2: Determine the y-intercept

The y-intercept (\(b\)) is the point where the line crosses the y-axis. From the slope-intercept equation, you can see that the y-intercept is:

- **Example**: For \(y = -\frac{2}{3}x + 2\), the y-intercept is 2. Plot the point (0, 2) on the graph.

### Step 3: Determine the slope

The slope (\(m\)) indicates how steep the line is and the direction it goes. It is expressed as a fraction \(\frac{\text{rise}}{\text{run}}\):

- **Example**: In the equation \(y = -\frac{2}{3}x + 2\), the slope is \(-\frac{2}{3}\). This means that for every 2 units you move down (rise: -2), you move 3 units to the right (run: +3).

### Step 4: Plot another point using the slope

Using the y-intercept, apply the slope to find another point:

- From (0, 2), go down 2 units (to y = 0) and right 3 units (to x = 3). This gives you a new point (3, 0).
- Plot this point on the graph.

### Step 5: Draw the line

Once you have at least two points plotted (the y-intercept and the second point), draw a straight line through them extending in both directions.

### Step 6: Label the graph (optional but helpful)

Mark your axes and indicate the linear equation of the line on the graph.

### Example

Let's graph the equation \(y = 2x - 1\):

1. **Identify**: The slope \(m = 2\) and y-intercept \(b = -1\).
2. **Y-intercept**: Plot (0, -1).
3. **Using slope**: From (0, -1), move up 2 units and right 1 unit to plot (1, 1).
4. **Draw the line**: Connect the points and extend the line.

You can plot more points for accuracy if needed.

### Conclusion

Graphing linear equations can help visualize relationships between variables and provide a better understanding of the equation's behavior. Would you like more examples or specific problems to solve?