Forms of Linear Equations
Linear equations in two variables x and y can be written in several forms. Common examples include:
Slope-intercept form: y = mx + c, where m is the slope and c is the y-intercept.
Rearranged forms like y = 7 - 4x, which can be rewritten as y = -4x + 7 to match the slope-intercept form for easier graphing.
- Equations may not always be given in y = mx + c form.
- Rearranging is often necessary to identify slope and intercept clearly.
Rearranging to y = mx + c
To graph a line easily, rewrite the equation so y is on one side by itself. For example, consider 3x + 2y = 5:
Subtract 3x from both sides: 2y = 5 - 3x, then divide both sides by 2: y = (5 - 3x)/2.
Simplify to y = -32x + 52, where the slope m = -32 and y-intercept c = 52.
- Isolate y by performing inverse operations.
- Express the equation clearly as y = mx + c for graphing.
Understanding Slope and Y-Intercept
The slope m indicates how steep the line is and the direction it goes:
A positive slope means the line rises as x increases.
A negative slope means the line falls as x increases. The slope is the ratio of vertical change (rise) to horizontal change (run). The y-intercept c is the point where the line crosses the y-axis (x=0). This is the starting point for plotting the graph.
- Slope m = change in y / change in x.
- Y-intercept c is the value of y when x = 0.
- Slope sign determines line direction.
Plotting the Graph
To draw the graph: first plot the y-intercept (0, c) on the coordinate axes.
Then use the slope m to find a second point: from the y-intercept, move vertically by the numerator of m and horizontally by the denominator (if m is a fraction). For example, if m = -2, move down 2 units and right 1 unit.
Draw a straight line through these points extending across the graph.
- Plot (0, c) first.
- Use slope to find a second point.
- Draw a straight line through the points.
Examples
1. y = -2x + 5: slope m = -2, y-intercept c = 5. Plot (0,5), then from there move down 2 units and right 1 unit to (1,3).
2. y = 7 - 4x: rewrite as y = -4x + 7. Slope m = -4, y-intercept c = 7. Plot (0,7), then move down 4 units and right 1 unit to (1,3).
3. 3x + 2y = 5: rearranged to y = (5 - 3x)/2 = -32x + 52. Plot (0, 2.5), then from there move down 3 units and right 2 units to (2, -0.5).
- Use rearrangement to find slope and intercept.
- Plot points carefully using slope and intercept.
- Draw the line through plotted points.
At-a-glance summary
Comparison table
| Step | Detail | Example | Exam Use |
|---|---|---|---|
| Identify equation form | Check if equation is y = mx + c or needs rearranging | 3x + 2y = 5 is not in y = mx + c | Always rearrange to y = mx + c for clarity |
| Rearrange equation | Isolate y on one side | y = (5 - 3x)/2 | Show clear algebraic steps for marks |
| Find slope and intercept | Slope m is coefficient of x; intercept c is constant term | y = -2x + 5 has m = -2, c = 5 | State slope and intercept explicitly |
| Plot y-intercept | Plot point (0, c) on y-axis | Plot (0,5) for y = -2x + 5 | Accurate plotting gains method marks |
| Use slope to find second point | From y-intercept, move rise/run according to m | From (0,5), move down 2, right 1 to (1,3) | Correct use of slope shows understanding |
| Draw line | Join points with a ruler to form straight line | Draw line through (0,5) and (1,3) | Neat line completes graph for full marks |
Active recall
Quick check
1. What is the slope and y-intercept of y = 7 - 4x?
Rewrite as y = -4x + 7; slope m = -4, y-intercept c = 7.
2. How do you find a second point after plotting the y-intercept?
Use the slope m = rise/run; from the y-intercept, move vertically by rise and horizontally by run to find the second point.
3. Rearrange 3x + 2y = 5 into y = mx + c form.
2y = 5 - 3x, so y = (5 - 3x)/2 = -32x + 52.
4. If the slope is -2, how do you move from the y-intercept to plot the next point?
Move down 2 units and right 1 unit from the y-intercept.