Section B — Chapters 23–39
Introduction to the Equation of a Line
Junior Cycle — 1st Year
- ✓By the end of this lesson students will be able to understand that a straight line can be represented by an algebraic equation.
- ✓By the end of this lesson students will be able to graph a straight line given its equation by creating a table of values and plotting points.
- ✓By the end of this lesson students will be able to identify the y-intercept of a line from its equation in the form y = mx + c.
- ✓By the end of this lesson students will be able to understand the meaning of 'm' (slope) in y = mx + c as a measure of steepness and direction.
- ✓By the end of this lesson students will be able to find the equation of a line in the form y = mx + c given its slope and a point it passes through.
Key concepts
An equation of a line is an algebraic rule that describes the relationship between the x-coordinates and y-coordinates of all the points that lie on that straight line. For example, in the equation y = x + 2, any point (x, y) that satisfies this rule (like (1, 3) or (0, 2)) lies on the line.
To graph a straight line from its equation, you can choose a few x-values, substitute them into the equation to find the corresponding y-values, and then plot these (x, y) co-ordinates on the co-ordinate plane. Since a straight line is determined by two points, finding at least two points is sufficient, but finding three points is a good way to check your work.
The equation of a straight line is often written in the form y = mx + c. In this form, 'm' represents the slope of the line, and 'c' represents the y-intercept. This is a very useful form because it tells us two key pieces of information about the line directly.
The slope, denoted by 'm', tells us how steep a line is and in which direction it goes. A positive slope (m > 0) means the line goes upwards from left to right. A negative slope (m < 0) means the line goes downwards from left to right. A larger absolute value of 'm' means the line is steeper.
The y-intercept, denoted by 'c', is the point where the line crosses the y-axis. At this point, the x-coordinate is always 0. So, the y-intercept is the point (0, c). You can find 'c' by setting x = 0 in the equation y = mx + c.
If you know the slope (m) of a line and one point (x₁, y₁) that it passes through, you can find its equation. You use the general form y = mx + c. Substitute the given slope 'm' and the co-ordinates (x₁, y₁) into the equation, and then solve for 'c'. Once 'c' is found, you can write the full equation of the line.
Key facts to remember
- 1A straight line can be represented by an algebraic equation that relates its x and y co-ordinates.
- 2To graph a line, find at least two points that satisfy its equation, plot them, and draw a straight line through them.
- 3The most common form for the equation of a straight line is y = mx + c.
- 4In y = mx + c, 'm' represents the slope, which indicates the steepness and direction of the line.
- 5In y = mx + c, 'c' represents the y-intercept, which is the point (0, c) where the line crosses the y-axis.
- 6If the slope 'm' is positive, the line goes up from left to right. If 'm' is negative, it goes down.
- 7To find the y-intercept (c) from an equation, set x = 0 and solve for y.
- 8To find the equation of a line given its slope and a point, substitute the values into y = mx + c to find 'c'.
Worked examples
Example 1
Graph the line with the equation y = 2x + 1 for x values from -2 to 2.
Answer
The graph is a straight line passing through the plotted points.
Always use a ruler to draw straight lines on the co-ordinate plane.
Example 2
For the line with equation y = -3x + 4, identify its slope and y-intercept.
Answer
The slope (m) is -3. The y-intercept (c) is 4, meaning the line crosses the y-axis at (0, 4).
A negative slope means the line goes downwards from left to right.
Example 3
Find the equation of a line that has a slope of 2 and passes through the point (1, 5).
Answer
The equation of the line is y = 2x + 3.
Always check your answer by substituting the given point back into your final equation.
Common mistakes
- ✗Incorrectly calculating y-values when substituting x-values into the equation, leading to wrong points.
- ✗Mixing up the x and y co-ordinates when plotting points (e.g., plotting (3, 2) instead of (2, 3)).
- ✗Not using a ruler to draw the line, resulting in a wobbly or inaccurate graph.
- ✗Confusing the slope 'm' with the y-intercept 'c' when reading an equation.
- ✗Not extending the line across the entire co-ordinate grid, making it look like a line segment rather than a continuous line.
Exam tips
- ★Always use a sharp pencil and a ruler when drawing graphs to ensure accuracy.
- ★Label your axes (x-axis, y-axis) and clearly mark the scale on both axes.
- ★Plot at least three points when graphing a line; if they don't lie on a straight line, you've made a mistake.
- ★Double-check your calculations for each point by substituting the co-ordinates back into the original equation.
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