Forms of Linear Equations
A straight line on the Cartesian plane can be represented by an equation involving x and y coordinates. Two widely used forms are:
Standard form: ax + by = c, where a, b, and c are constants and a and b are not both zero.
Slope-intercept form: y = mx + c, where m is the gradient (slope) of the line and c is the y-intercept (the point where the line crosses the y-axis).
- Standard form is useful for general representation and can describe vertical lines (when b=0).
- Slope-intercept form directly shows how y changes with x and the point where the line crosses the y-axis.
Deriving Gradient and Y-Intercept from Standard Form
To find the gradient and y-intercept from an equation in standard form, rearrange it into slope-intercept form by solving for y. This process involves isolating y on one side:
1. Start with ax + by = c (a, b, c are constants).
2. Subtract ax from both sides: by = -ax + c. 3. Divide every term by b: y = (-a/b)x + (c/b).
- The coefficient of x, -a/b, is the gradient m.
- The constant term, c/b, is the y-intercept.
Example: Finding Gradient and Y-Intercept
Consider the equation 5x + 4y = 8. To find the gradient and y-intercept:
Rearrange to solve for y: 4y = -5x + 8 Divide both sides by 4: y = (-5/4)x + 2
- Gradient m = -5/4.
- Y-intercept c = 2, meaning the line crosses the y-axis at (0, 2).
Determining Gradient and Y-Intercept from a Graph
From a graph, the gradient (m) is the ratio of the vertical change (rise) to the horizontal change (run) between any two distinct points on the line:
\[ m = \frac{\text{rise}}{\text{run}} = \frac{y_2 - y_1}{x_2 - x_1} \]
The y-intercept is the y-coordinate where the line crosses the y-axis (where x = 0). This point is written as (0, c).
- Choose two clear points on the line to calculate the gradient.
- Locate the point where the line meets the y-axis to find the y-intercept.
Simplifying Equations Fully
Always express the equation of a line in its simplest form:
Coefficients should be integers with no common factors (unless fractions are preferred).
The equation should be rearranged neatly, typically with y isolated on one side for slope-intercept form or with integer coefficients in standard form.
- Simplification improves clarity and reduces errors in interpretation.
- For example, 10x + 8y = 16 can be simplified by dividing all terms by 2 to get 5x + 4y = 8.
At-a-glance summary
Comparison table
| Step | Detail | Example | Exam Use |
|---|---|---|---|
| Identify form | Check if equation is standard (ax + by = c) or slope-intercept (y = mx + c) | 5x + 4y = 8 is standard form | Recognise form to decide rearrangement method |
| Rearrange | Solve for y to get slope-intercept form if needed | From 5x + 4y = 8, rearranged to y = (-5/4)x + 2 | Find gradient and y-intercept clearly |
| Calculate gradient | Gradient m = coefficient of x in slope-intercept form | m = -5/4 | State gradient precisely for marks |
| Find y-intercept | Y-intercept c = constant term in slope-intercept form | c = 2 | Identify point where line crosses y-axis |
| From graph | Calculate rise/run between two points; read y-intercept at x=0 | Points (2, -0.5) and (0, 2) give m = (-0.5 - 2)/(2 - 0) = -2.5/2 = -1.25 | Use graph to verify or find equation components |
| Simplify | Reduce coefficients to simplest integer ratio or simplest fraction | 10x + 8y = 16 simplified to 5x + 4y = 8 | Avoid losing marks for unsimplified answers |
Active recall
Quick check
1. Given the equation 3x + 6y = 12, what is the gradient and y-intercept?
Rearrange: 6y = -3x + 12, so y = (-3/6)x + 12/6 = (-1/2)x + 2. Gradient m = -1/2, y-intercept c = 2.
2. From a graph, two points on a line are (1, 3) and (4, 9). What is the gradient?
Gradient m = (9 - 3) / (4 - 1) = 6 / 3 = 2.
3. If a line crosses the y-axis at (0, -4), what is the y-intercept?
The y-intercept is -4.
4. Simplify the equation 8x + 12y = 20 before finding the gradient.
Divide all terms by 4: 2x + 3y = 5. Then rearrange to y = (-2/3)x + 5/3.