MPM2D Solving Linear Systems by Graphing: Notes and Practice

You and a friend want to rent a canoe. Riverside Rentals charges $15 to start plus $10 for every hour. Lakeview Rentals charges $35 to start plus $5 for every hour. Riverside is cheaper for a quick paddle, because its starting fee is lower. Lakeview is cheaper for a long day, because its hourly rate is lower. So somewhere in between there must be a number of hours where the two companies charge exactly the same amount. If $h$ is the number of hours and $C$ is the cost in dollars, Riverside's cost is $C = 10 h + 15$ and Lakeview's cost is $C = 5 h + 35$ . Finding the one value of $h$ that gives both equations the same $C$ is called solving a linear system. In this lesson you will solve systems by drawing both lines and finding where they cross. In the next two lessons you will learn algebra methods that give exact answers every time.

A linear equation in two variables is an equation like $y = 2 x - 1$ or $3 x + 4 y = 12$ : each variable appears only to the first power, and its graph is a straight line. A linear system is a set of two (or more) linear equations in the same variables that must be true at the same time. A solution of the system is an ordered pair $(x, y)$ that makes every equation in the system true. Every point on a line makes that line's equation true, and no other point does. So a point that satisfies both equations has to lie on both lines: it is the point of intersection, where the lines cross. A brace is often used to show that equations belong together:

{C = 10 h + 15 C = 5 h + 35

Before you solve anything, learn to check. To test whether an ordered pair is a solution, substitute its coordinates into each equation, one at a time. It is a solution only if both sides match in every equation. Test $(4, 55)$ in the canoe system: Riverside gives $10 (4) + 15 = 40 + 15 = 55$ , which matches $C = 55$ . Lakeview gives $5 (4) + 35 = 20 + 35 = 55$ , which also matches. So $(4, 55)$ is the solution: for a 4-hour trip, both companies charge $55. Now test $(2, 35)$ : Riverside gives $10 (2) + 15 = 35$ , a match, but Lakeview gives $5 (2) + 35 = 45$ , not $35$ . The point $(2, 35)$ lies on the Riverside line only, so it is not a solution of the system.

Both equations, every time

A point that works in only one equation is just a point on one of the lines. The solution of a system must make every equation true. Always check in the original equations, not in ones you rearranged, so a rearranging slip cannot hide.

Rows of red, yellow and blue rental canoes and kayaks stacked on the grass beside a lake, with a small rental shed behind them. — PhotoRental companies set prices with a starting fee plus an hourly rate, so comparing two of them is a linear system whose solution is the trip length where both cost the same.Thomson200, CC0, via Wikimedia Commons

Solving by graphing

Write each equation in slope and y-intercept form $y = m x + b$ , or find its two intercepts. Here $m$ is the slope (rise over run) and $b$ is the y-intercept (where the line crosses the y-axis).
Graph both lines carefully on the same set of axes, using a ruler and a scale that shows where they meet.
Read the coordinates of the point of intersection.
Check the point by substituting it into both original equations, then state the solution as an ordered pair.

Many equations are given in standard form, $A x + B y = C$ , with the $x$ and $y$ terms on the same side. You can graph a standard-form line in two ways. Use the intercepts: for $2 x + 3 y = 12$ , set $x = 0$ to get $3 y = 12$ , so $y = 4$ ; set $y = 0$ to get $2 x = 12$ , so $x = 6$ . Plot $(0, 4)$ and $(6, 0)$ and join them. Or rearrange into $y = m x + b$ : subtract $2 x$ from both sides to get $3 y = - 2 x + 12$ , then divide every term by 3 to get $y = - \frac{2}{3} x + 4$ . Both routes give the same line. The intercept method is quickest when the intercepts are whole numbers. Rearranging is better when you want the slope, for example to compare two lines.

Fig 1Left: the lines y = 2x − 1 and x + y = 5 cross at (2, 3), the one solution of the system. Right: the three ways two lines can meet, with one solution, no solution, or infinitely many solutions.

VideoKhan Academy graphs two lines on the same axes and reads their point of intersection as the solution of the system.Watch on YouTube

One solution, no solution or infinitely many

Two straight lines in the same plane can meet in only three ways, and comparing their slopes and y-intercepts tells you which one you have before you draw anything. If the slopes are different, the lines cross at exactly one point, so the system has one solution. If the slopes are the same but the y-intercepts are different, the lines are parallel: they never meet, so the system has no solution. If the slopes and the y-intercepts are both the same, the two equations describe the same line (the lines are called coincident), so every point on that line is a solution and there are infinitely many solutions. A system with at least one solution is called consistent; a system with no solution is inconsistent. One special case: a vertical line such as $x = 3$ has no slope you can compare, so think about the picture instead. Two different vertical lines are parallel, and a vertical line always crosses a non-vertical line exactly once.

Slopes	y-intercepts	Graph	Number of solutions
Different	Same or different	Lines cross once	Exactly one
Same	Different	Parallel lines, never meet	None
Same	Same	One line drawn twice (coincident)	Infinitely many

Comparing two non-vertical lines written as y = mx + b

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Enter two lines in slope and y-intercept form or standard form, or drag them by their points, and watch where they cross. Try parallel lines and the same line twice.

The limits of graphing

Graphing shows you the whole situation at a glance, but reading a point off a graph is only as exact as your drawing. Solve $y = x + 1$ and $y = - 2 x + 3$ by graphing, and the lines seem to cross near $(0.7, 1.7)$ . Check it: in the first equation $0.7 + 1 = 1.7$ , which works, but in the second $- 2 (0.7) + 3 = 1.6$ , not $1.7$ . The exact solution is $(\frac{2}{3}, \frac{5}{3})$ , which no hand-drawn graph can show precisely. Graphing technology (a graphing calculator or graphing software) can locate the crossing to several decimal places, but it still gives a rounded decimal. When a solution is not a pair of whole numbers, or the numbers are too large to fit on a grid, use substitution or elimination instead.

Use graphing to estimate and to see

Even when you solve with algebra, a quick sketch tells you roughly where the answer should be and whether the system has one, no or infinitely many solutions. If your algebra gives

x = 40

but the sketch shows the lines crossing near

x = 4

, look for an error.

Solving Linear Systems by Graphing

Key things to remember

Solving by graphing

One solution, no solution or infinitely many

The limits of graphing

Worked examples

Where marks get lost

Flashcards 9

Practice