Plot the linear equation y = 2x - 3 and find x and y intercepts

Question

For the linear equation $y = 2x - 3$ : (a) Find the $x$ -intercept and $y$ -intercept. (b) Plot the graph on the Cartesian plane. (c) State the slope and interpret it.

Solution — Step by Step

The $y$ -intercept is where the line crosses the $y$ -axis — at this point, $x = 0$ .

y = 2(0) - 3 = -3

$y$ -intercept: $(0, -3)$ — the line crosses the $y$ -axis at $y = -3$ .

In the equation $y = mx + c$ (slope-intercept form), the $y$ -intercept is always the constant $c$ . Here, $c = -3$ .

The $x$ -intercept is where the line crosses the $x$ -axis — at this point, $y = 0$ .

0 = 2x - 3 \implies 2x = 3 \implies x = \frac{3}{2} = 1.5

$x$ -intercept: $(1.5, 0)$ — the line crosses the $x$ -axis at $x = 1.5$ .

Choose any value, e.g., $x = 2$ :

y = 2(2) - 3 = 4 - 3 = 1

Third point: $(2, 1)$ .

We now have three points: $(0, -3)$ , $(1.5, 0)$ , and $(2, 1)$ . Plot these and draw a straight line through them. All three should be collinear (lying on the same line) — if they’re not, recheck your arithmetic.

From $y = 2x - 3$ , the slope $m = 2$ .

Interpretation: For every 1 unit increase in $x$ , $y$ increases by 2 units. The slope is positive, so the line goes upward from left to right. A slope of 2 is quite steep — it makes an angle of $\tan^{-1}(2) \approx 63°$ with the $x$ -axis.

Why This Works

The equation $y = mx + c$ (slope-intercept form) directly tells us two key features:

$m$ (coefficient of $x$ ) = slope — the rate of change of $y$ with respect to $x$
$c$ (constant term) = $y$ -intercept — the value of $y$ when $x = 0$

To plot any linear equation, two points are mathematically sufficient (two points determine a line). We find three points as a safety check — if all three align, we’ve made no error.

Alternative Method

Convert to standard form $ax + by + c = 0$ : $2x - y - 3 = 0$ .

$x$ -intercept: set $y = 0$ → $2x - 3 = 0$ → $x = 3/2$ ✓

$y$ -intercept: set $x = 0$ → $-y - 3 = 0$ → $y = -3$ ✓

Both methods give the same intercepts.

The fastest way to plot any line: find the two intercepts (set $x=0$ for $y$ -intercept; set $y=0$ for $x$ -intercept), plot both points, and draw the line through them. For the equation $y = 2x - 3$ : two points $(0, -3)$ and $(1.5, 0)$ are enough — no need to find more points unless you want a verification check.

Common Mistake

Students sometimes confuse the $x$ and $y$ intercepts — finding where the line crosses the $y$ -axis but calling it the “ $x$ -intercept.” Remember: the $y$ -intercept is the point where you cross the $y$ -axis (so $x = 0$ there). The $x$ -intercept is the point where you cross the $x$ -axis (so $y = 0$ there). Another common error: plotting $x$ -intercept $(1.5, 0)$ but writing the answer as $(0, 1.5)$ — the intercept point is always $(x_0, 0)$ for $x$ -intercept and $(0, y_0)$ for $y$ -intercept, never the other way around.

Question

Solution — Step by Step

Why This Works

Alternative Method

Common Mistake

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