Mathematics · Ch 10 — Straight Lines

Slope-intercept Form

10.3.4

Slope-intercept Form

Slope-Intercept Form

When a line's slope is known and we also know where it cuts one of the coordinate axes, we can write its equation directly. The intercept — the signed distance from the origin to the point where the line meets an axis — gives us a fixed point on the line, and the slope gives us the direction. Together, these two pieces of information determine the line uniquely.

Case I: Line with given slope and y-intercept

Suppose a line $L$ has slope $m$ and cuts the $y$ -axis at a distance $c$ from the origin. This distance $c$ is called the $y$ -intercept of the line. The point where the line meets the $y$ -axis has coordinates $(0, c)$ .

Since the line has slope $m$ and passes through the fixed point $(0, c)$ , we can use the point-slope form. The point-slope equation of a line through $(x_1, y_1)$ with slope $m$ is $y - y_1 = m(x - x_1)$ . Substituting $x_1 = 0$ and $y_1 = c$ :

y - c = m(x - 0)

This simplifies to:

y - c = mx

y = mx + c

Thus, a point $(x, y)$ lies on the line with slope $m$ and $y$ -intercept $c$ if and only if

y = mx + c

The value of $c$ is positive when the intercept is on the positive side of the $y$ -axis (above the origin) and negative when the intercept is on the negative side (below the origin).

Watch out

Do not confuse the $y$ -intercept $c$ with the $y$ -coordinate of a general point on the line. The intercept is a fixed number — the $y$ -coordinate of the specific point where the line crosses the $y$ -axis. Every other point on the line has a different $y$ -coordinate.

Case II: Line with given slope and x-intercept

Now suppose a line $L$ has slope $m$ and makes an $x$ -intercept $d$ . This means the line cuts the $x$ -axis at the point $(d, 0)$ . The distance $d$ is called the $x$ -intercept of the line.

Using the same method as in Case I, we apply the point-slope form with the fixed point $(d, 0)$ :

y - 0 = m(x - d)

This gives:

y = m(x - d)

Tip

You can derive this equation yourself by exactly the same reasoning as in Case I — just replace the fixed point $(0, c)$ with $(d, 0)$ . The structure is identical: slope $m$ times $(x - \text{intercept coordinate})$ .

Worked Example

Example 7. Write the equation of the lines for which $\tan \theta = \frac{1}{2}$ , where $\theta$ is the inclination of the line, and:

(i) $y$ -intercept is $-\frac{2}{3}$

(ii) $x$ -intercept is $4$

Solution.

(i) Here, the slope of the line is $m = \tan \theta = \frac{1}{2}$ , and the $y$ -intercept is $c = -\frac{2}{3}$ .

Using the slope-intercept form $y = mx + c$ :

y = \frac{1}{2}x + \left(-\frac{2}{3}\right)

y = \frac{1}{2}x - \frac{2}{3}

Multiplying through by 6 to clear denominators:

6y = 3x - 4

Rearranging to standard form:

3x - 6y - 4 = 0

This is the required equation.

(ii) Here, $m = \frac{1}{2}$ and the $x$ -intercept is $d = 4$ . …

Figure 9.12Slope-intercept form

Drawn by us to help you understand the concept clearly, and verified to make sure it's accurate. For exams, practice from your textbook's own diagram.

Fig. 9.12 is a simple coordinate-plane sketch that shows the geometric meaning of the slope-intercept form of a straight line. The axes are the usual $x$ -axis (horizontal) and $y$ -axis (vertical). A single straight line, labelled $L$ , is drawn sloping upward from left to right — its slope is $m$ , and the value $m$ is written along the line itself, tilted to follow its direction. The line crosses the $y$ -axis at a single marked point with coordinates $(0, c)$ . That crossing point is the $y$ -intercept; the distance $c$ is measured from the origin $(0,0)$ along the $y$ -axis, and $c$ can be positive (above the origin) or negative (below it). No other curves, points, or labels appear in the figure — it is deliberately minimal, focusing attention on just two pieces of information: the slope and the $y$ -intercept.

The physical idea is this: if you know how steep a line is (its slope $m$ ) and exactly where it meets the $y$ -axis (the intercept $c$ ), then you can write its equation immediately — you don’t need any other point. The figure makes that relationship visual: the line is completely determined by those two numbers.

From this diagram, the textbook derives the central formula of the section. Since the line has slope $m$ and passes through the fixed point $(0, c)$ , the point-slope form gives:

y - c = m(x - 0)

which simplifies to:

y = mx + c

Here, $m$ is the slope of the line, and $c$ is the $y$ -intercept — the $y$ -coordinate of the point where the line meets the $y$ -axis. Every point $(x, y)$ on the line satisfies this equation, and conversely any point satisfying it lies on the line. The sign of $c$ tells you whether the intercept lies on the positive or negative side of the $y$ -axis.

Watch out

Do not confuse the $y$ -intercept $c$ with the $x$ -intercept. The $y$ -intercept is the value of $y$ when $x = 0$ ; the $x$ -intercept is the value of $x$ when $y = 0$ . The figure shows only the $y$ -intercept case. The textbook separately treats the $x$ -intercept case (Case II), which gives $y = m(x - d)$ , where $d$ is the $x$ -intercept. …

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