Draw the lines, then read the answer off the point where they cross

How to use this simulation

1. Look before you touch anything. In "One line · y = mx + c" mode the screen shows y = 2x − 4. It crosses the x-axis at (2, 0) and the y-axis at (0, −4), with an angle of about 63.43°.
2. Move the slope slider m: the line gets steeper or flatter, and for a negative m it falls to the right. The c slider slides the whole line up or down without changing its slope.
3. Switch to "One line · two points" and drag the two points. The slope updates live. Put both points at the same x and the line turns vertical, with the slope reported as undefined.
4. Now "Two lines · simultaneous equations". It starts with 2x + 3y = 12 and x − y = 1. The orange dot at (3, 2) is the solution, and the readings panel says "One solution".
5. Drag a₂ and b₂ until the second line is parallel to the first: the crossing vanishes and the case turns into "No solution". Then adjust c₂ until the lines overlap and the case becomes "Infinitely many solutions".
6. Keep "Show the working step by step" on. The slope, intercepts, angle, ratio test and cross-multiplication are worked out below with your own numbers. Use "Visible range" to zoom out and catch far-away intersections.

Two rides, one distance where the fares are equal

Two rickshaw drivers wait outside your house. The first says, "Pay 20 up front, then 10 per kilometre." The second says, "No up-front fee, 15 per kilometre." Whose rickshaw should you take?

It depends on how far you are going. For a short trip the second driver is cheaper, because the first driver's up-front fee hurts. For a long trip the first driver wins, because his rate per kilometre is lower. Somewhere in between there is a distance where both charge exactly the same. To find it, call the fare y and the distance x, and write two equations: y = 10x + 20 and y = 15x.

Both are linear, so each graphs as a straight line. Where the lines cross, one x gives the same y on both, which means the same fare. Working it out gives 4 km, where both drivers charge 60. Below that distance the second driver is cheaper, and above it the first.

This page is about finding that crossing point. In the simulation above you will draw a line, understand slope and intercepts, then put two equations on the same grid and read the answer. You will also see why two lines sometimes never meet and sometimes lie on top of each other.

From scratch: the coordinate plane

Draw a horizontal line and a vertical line on paper. The point where they cross is the origin, written O. The horizontal line is the x-axis and the vertical one is the y-axis. Numbers go positive to the right and up, and negative to the left and down.

Any point in the plane is named by two numbers (x, y). The first says how far right (or left) of the origin, the second how far up (or down). This is an ordered pair. (3, 2) means 3 steps right and 2 up, but (2, 3) is a different point, so the order matters.

The axes cut the plane into four quadrants. In the first both x and y are positive; in the second x is negative and y positive; in the third both are negative; in the fourth x is positive and y negative. Every point on the x-axis has y = 0, and every point on the y-axis has x = 0. You will use those two facts in a minute to find intercepts.

The grid in the simulation is exactly this: the origin in the middle and one unit per square. The "Visible range" slider zooms the picture, so you can read small numbers on big squares or catch a distant intersection on small ones.

A linear equation is a straight line

The equation ax + by = c (with a and b not both zero) has endless solutions. For x + y = 5 you get (1, 4), (2, 3), (5, 0), (7, −2), even (0.5, 4.5). Pick any x and y follows.

Plot those solutions and they all land on one straight line. The reverse also holds: every point on that line satisfies the equation. So the equation and the line are two ways of writing the same thing, which is why we call it a linear equation.

That gives a handy result. You need only two points to draw the line, because two different points fix exactly one straight line. A third point is a good habit as a check: if the three do not line up, an arithmetic slip hides somewhere.

Two equations taken together are a system, also called simultaneous equations. A solution of the system is an (x, y) that satisfies both. On the graph it is a point lying on both lines, and a point on two lines can only be where they cross.

The words you need, in one place

Knowledge-type exam questions come straight from these terms, so keep the table handy.

Slope: read it from two points

Slope measures steepness. Think of a staircase: how far forward you go on each step is the run, and how far up is the rise. Slope = rise ÷ run. The dashed grey triangle in the simulation shows exactly these two lengths.

For any two points (x₁, y₁) and (x₂, y₂) on a line, m = (y₂ − y₁) / (x₂ − x₁). With the starting points (−2, −3) and (2, 5), the rise is 5 − (−3) = 8, the run is 2 − (−2) = 4, and so m = 8/4 = 2.

Swapping the two points does not change the slope, since top and bottom both flip sign. The only rule is to subtract in the same order on top and bottom. Write (y₂ − y₁) over (x₁ − x₂) and the sign will come out wrong.

The sign of the slope tells you the direction. The bigger the slope, the steeper the line. A slope of zero is a flat line. If the run is zero you cannot divide, so a vertical line has an undefined slope. Put both points at the same x in the simulation and watch it happen.

Slope also links to angle. If the line makes an angle θ with the positive x-axis, then tan θ is the slope. A slope of 1 means θ = 45°. The default line has m = 2, so θ ≈ 63.43°. A negative slope gives a negative angle: the line falls to the right.

Forms of the equation of a line

The same line can be written several ways, and much of the work is moving between them. The one used most is slope-intercept form, y = mx + c. Here m is the slope and c is the y-intercept: the y you get at x = 0. The line meets the y-axis at (0, c).

Pairs of equations usually arrive in standard form ax + by = c. Solving for y gives y = (−a/b)x + c/b, so the slope is −a/b and the y-intercept is c/b. For the x-intercept put y = 0 and get x = c/a. For the first starting equation 2x + 3y = 12, the slope is −2/3 = −2/3, and the intercepts are (6, 0) and (0, 4).

Intercept form, x/p + y/q = 1, is handy for drawing. The line crosses the x-axis at (p, 0) and the y-axis at (0, q). Dividing the standard form by c produces it. You can sketch the line almost in your head: mark the two axis points and join them.

If you know the slope and one point (x₀, y₀), use point-slope form, y − y₀ = m(x − x₀). With two points, first work out the slope and then use this form. A vertical line has its own form, x = k, with no slope; it cannot be written as y = mx + c, so whenever b = 0 think separately.

Solving a pair by graphing, step by step

Suppose we must solve 2x + 3y = 12 and x − y = 1. The steps never change, so learn them once.

• Make a table of values for each equation. Choose x values that make y a whole number.
• Draw the axes on graph paper and pick a comfortable scale, for example one small square per unit.
• Plot the points of the first equation, join them with a ruler and extend the line both ways.
• On the same axes, plot the second equation and draw its line.
• Find where the lines cross and read off the coordinates of that point. That is the solution.
• Substitute into both equations to check.

Three outcomes and the ratio test

Two lines in a plane can sit in only three ways, and those are the three outcomes for a system. (1) They cross at one point: one solution, and the system is consistent and independent. (2) They are parallel and never meet: no solution, and the system is inconsistent. (3) They are the same line: infinitely many solutions, and the system is consistent but dependent.

You can tell which without drawing, by comparing the coefficients. For a₁x + b₁y = c₁ and a₂x + b₂y = c₂, compare a₁/a₂, b₁/b₂ and c₁/c₂. If a₁/a₂ ≠ b₁/b₂ the slopes differ, so the lines must cross. If a₁/a₂ = b₁/b₂ the slopes are equal, so the lines are parallel or identical. Then c₁/c₂ decides: different means parallel, equal means the same line.

In the starting pair a₁/a₂ = 2 and b₁/b₂ = −3. They differ, so the lines cross. The readings panel in the simulation always shows the three ratios and a₁b₂ − a₂b₁ (here −5). A determinant that is not zero guarantees exactly one solution.

The algebraic version is cross-multiplication: when D = a₁b₂ − a₂b₁ ≠ 0, x = (c₁b₂ − c₂b₁) / D and y = (a₁c₂ − a₂c₁) / D. For the starting pair, D = −5, x = −15/(−5) = 3 and y = −10/(−5) = 2. That matches the graph.

Try it in the simulation

Each experiment takes a minute or two. Guess first, then check.

Solved examples (every number computed)

Each problem follows the steps you would write in an exam. Example 1 is the simulation's starting line and examples 2 and 3 are systems.

Mistakes everyone makes

Each of these costs marks, so watch for them.

• Writing an ordered pair backwards: (2, 3) and (3, 2) are different points, and x comes first.
• Mixing the order of the points in the slope formula. The point that is first on top must be first on the bottom.
• Taking the slope of ax + by = c as a/b. It is −a/b, and the sign matters.
• Writing "infinitely many" for parallel lines, or forgetting to look at the third ratio when the first two are equal.
• Putting x = 0 to find the x-intercept. For the x-intercept, set y = 0.
• Reading a solution off the graph and not substituting it into both equations.
• Using different scales on the two axes, which distorts the slope.
• Drawing a short segment instead of extending the line both ways, which can hide the crossing.
• Forcing a vertical line into y = mx + c. It has no slope; its equation is x = k.

Linear equations in real life

Graphs and intersection points sit behind many everyday decisions:

• Choosing a phone plan: one has a higher monthly fee and a lower rate per minute, the other the reverse. The crossing tells you which is cheaper for how you use it.
• Ride-hailing and taxi fares: a booking fee plus a rate per kilometre makes a straight line.
• Business break-even: the revenue line and the cost line cross where profit begins.
• Steepness of a road or ramp: a larger slope means a harder climb, which is why wheelchair ramps are kept gentle.
• Motion: on a distance versus time graph, slope is speed, and two lines meeting means two travellers are at the same place at the same time.
• Phone battery against hours used: a straight-line trend lets you estimate how long it will last.

Exam corner

Questions on this topic usually give two equations or a short situation. Part (a) tests a definition or a quick substitution, such as the slope or whether a point satisfies an equation. Part (b) asks you to solve the pair, or to find k from a condition on the ratios.

Part (c) generally asks for a graph and the solution read from the intersection. Part (d) may ask for the area of the triangle the line cuts off with the axes, or to explain a result. When graphing, state the scale, label the axes, mark and name the points.

For multiple choice remember: a₁/a₂ ≠ b₁/b₂ gives one solution; all three ratios equal gives infinitely many; first two equal and the third different gives none. For ax + by = c the slope is −a/b, the x-intercept c/a and the y-intercept c/b. A vertical line has an undefined slope and a horizontal line has slope 0.

Revision on one page

The night before the exam, just run through this:

• The graph of ax + by = c is a straight line. Two points draw it; a third checks it.
• Slope m = (y₂ − y₁)/(x₂ − x₁) = tan θ = −a/b.
• In y = mx + c, m is the slope and c the y-intercept. For the x-intercept put y = 0.
• The solution of a pair of equations is the point where the two lines meet.
• a₁/a₂ ≠ b₁/b₂: one solution, consistent, independent.
• a₁/a₂ = b₁/b₂ ≠ c₁/c₂: parallel, no solution, inconsistent.
• a₁/a₂ = b₁/b₂ = c₁/c₂: same line, infinitely many solutions, dependent.
• Cross-multiplication: x = (c₁b₂ − c₂b₁)/D, y = (a₁c₂ − a₂c₁)/D, D = a₁b₂ − a₂b₁.
• A vertical line x = k has an undefined slope. A horizontal line y = k has slope zero.

Frequently asked questions