Analytic Geometry · Analytic Geometry

Linear Systems: Solving by Graphing

1.2About 70 minutes

Learning goals

I can explain what a linear system is and what its solution means.
I can check whether an ordered pair is a solution of a linear system.
I can solve a linear system by graphing and interpret the point of intersection.
I can tell whether a system has one solution, no solution or infinitely many solutions.

Words to know

linear system
Two or more linear equations in the same variables, considered together.

solution of a system
An ordered pair that makes every equation in the system true.

point of intersection
The point where two lines cross; it lies on both lines.

parallel lines
Lines in the same plane with equal slopes that never meet.

coincident lines
Two equations that describe the same line.

standard form
A linear equation written as ax + by = c (or Ax + By + C = 0).

Two lines, one question

Your family needs a driveway plowed this winter. Clearway charges $20 per plow. Snowbusters charges a $60 season fee plus $10 per plow. Which is the better deal?

Each company's price is a linear relation. Clearway: C = 20n. Snowbusters: C = 10n + 60. The question "when do they cost the same?" asks for an ordered pair (n, C) that makes both equations true at once. Two equations that must be true at the same time are called a linear system. Solving linear systems is the first big idea of this unit.

Remember from Grade 9

In Grade 9 you graphed lines in the forms y = mx + b and ax + by = k, and you found the point of intersection of two linear relations by graphing them and by setting the two expressions equal. You also learned that lines with the same slope are parallel. This lesson builds on that. Over the next three lessons you will add two algebraic methods, substitution and elimination, that work for any linear system.

What is a linear system?

A linear system (or system of linear equations) is a set of two or more linear equations that use the same variables. In this course, systems have two equations and two variables, usually x and y. For example:

x + y = 6

y = 2x − 3

A solution of the system is an ordered pair (x, y) that makes both equations true. On a graph, every point on a line makes that line's equation true. So the point that is on both lines, the point of intersection, is the solution.

How to check a solution

Substitute the x- and y-values into each equation. The left side must equal the right side in both. If even one equation fails, the ordered pair is not a solution.

Worked example

Example 1: Is it a solution?

Is (2, −1) a solution of the system 3x + 2y = 4 and x − y = 3? Is (0, 2) a solution?

Check (2, −1) in the first equation: left side 3(2) + 2(−1) = 6 − 2 = 4. Right side 4. ✓
Check (2, −1) in the second equation: left side 2 − (−1) = 2 + 1 = 3. Right side 3. ✓
Both equations are true, so (2, −1) is the solution.
Check (0, 2) in the first equation: 3(0) + 2(2) = 4 ✓. In the second: 0 − 2 = −2, but the right side is 3. ✗
(0, 2) is on the first line only, so it is not a solution of the system.

(2, −1) is a solution; (0, 2) is not, because it fails the second equation.

Solving by graphing

Graph both lines on the same grid. Use whichever method is quickest for each equation:

y = mx + b form: plot the y-intercept b, then use the slope m (rise over run) to find more points.
Standard form ax + by = c: find the two intercepts. Put x = 0 to find the y-intercept, and y = 0 to find the x-intercept.

Find where the lines cross. Read the coordinates carefully.
Check the point in both original equations.

Use a ruler and a sharp pencil. A small error in drawing can move the crossing point.

Worked example

Example 2: Solve by graphing

Solve by graphing: x + y = 6 and y = 2x − 3.

Line 1, x + y = 6 (standard form). If x = 0, then y = 6, so the y-intercept is (0, 6). If y = 0, then x = 6, so the x-intercept is (6, 0). Draw the line through these two points.
Line 2, y = 2x − 3 (slope-intercept form). The y-intercept is (0, −3). The slope is 2 = 2/1: from (0, −3) go right 1 and up 2 to (1, −1), then again to (2, 1), (3, 3) and so on.
The lines cross at (3, 3).
Check in line 1: 3 + 3 = 6 ✓. Check in line 2: 2(3) − 3 = 6 − 3 = 3 ✓.

The solution is (3, 3), that is, x = 3 and y = 3.

The solid line is x + y = 6 (through its intercepts (0, 6) and (6, 0)). The dashed line is y = 2x − 3 (y-intercept −3, slope 2). They cross at the solution (3, 3).

One, none or infinitely many

Two lines in a plane can meet in three ways, so a linear system can have three kinds of answers.

One solution. The lines have different slopes, so they cross exactly once.
No solution. The lines have the same slope but different y-intercepts. They are parallel and never meet.
Infinitely many solutions. The two equations describe the same line (same slope and same y-intercept). Every point on the line is a solution. Lines like this are called coincident.

For example, 2x + y = 3 and 4x + 2y = 6 are the same line, because multiplying the first equation by 2 gives the second.

The three possibilities for two lines. Left: different slopes, one crossing point. Middle: same slope, different y-intercepts, no crossing. Right: the dashed line lies exactly on top of the solid line, so every point is shared.

Worked example

Example 3: The plow problem

Clearway charges C = 20n and Snowbusters charges C = 10n + 60, where n is the number of plows and C is the total cost in dollars. Solve the system by graphing and decide which company is cheaper.

Graph C = 20n: it starts at (0, 0) and rises $20 per plow: (1, 20), (2, 40), ... (6, 120).
Graph C = 10n + 60: it starts at (0, 60) and rises $10 per plow: (1, 70), (2, 80), ... (6, 120).
The lines cross at (6, 120). Check: 20(6) = 120 ✓ and 10(6) + 60 = 120 ✓.
Interpret: for 6 plows, both companies charge $120.
To the left of n = 6, the Clearway line is lower, so Clearway is cheaper. To the right of n = 6, the Snowbusters line is lower. For example, for 10 plows: Clearway $200, Snowbusters $160.

Choose Clearway for fewer than 6 plows, Snowbusters for more than 6. At exactly 6 plows they cost the same, $120. In a snowy winter with many storms, Snowbusters wins.

Both plans cost $120 for 6 plows. To the left of the crossing point the solid Clearway line is lower (cheaper). To the right the dashed Snowbusters line is lower.

Limits of graphing

Graphing shows the whole situation at a glance, and it is great for comparing plans. But it can be inaccurate when the crossing point is not on a grid point, for example (2.4, −1.75). Graphing technology such as Desmos or GeoGebra (both free) can find the point exactly: graph both lines and tap the intersection. In the next two lessons you will learn algebraic methods that always give the exact answer.

Just for fun

Parallel lines have so much in common. It's a shame they'll never meet.

Hands-on activity

String-and-grid systems

You need: graph paper (or a grid drawn with a ruler), a ruler, a pencil and two pieces of string or thread about 30 cm long, and some tape.

Draw x- and y-axes from −8 to 8 on the graph paper.
Graph y = x + 1 using the string: tape one end at (0, 1) and stretch it through (3, 4). Tape the other end down.
Graph x + 2y = 8 with the second string: find its intercepts (8, 0) and (0, 4), and tape the string through them.
Read the crossing point. Then check it in both equations.
Now slide the second string, keeping it the same steepness, so it passes through (0, 2) instead. Where do the strings cross now? Check your new point in x + 2y = 4.
Finally, place the second string so it never crosses the first. What must be true about its steepness?

What to look for: step 4 gives (2, 3), since 2 + 1 = 3 and 2 + 2(3) = 8. Step 5 gives (2/3, 5/3), about (0.67, 1.67): hard to read from a string, which shows why exact algebra methods are useful. Step 6: the strings never cross when they are parallel (slope 1, like y = x + 1).

Your turn

Is (4, −1) a solution of x + y = 3 and 2x − y = 9?

Is (1, 5) a solution of y = 3x + 2 and x + y = 7?

Solve by graphing: y = x + 2 and y = −x + 4.

Solve by graphing: 2x + y = 8 and y = x − 1.

Find the x- and y-intercepts of 3x − 2y = 12.

Without graphing, how many solutions does y = 4x − 1 and y = 4x + 5 have? Explain.

A gym charges $30 per month plus $5 per visit. A community centre charges $10 per visit. Write the two equations, solve by graphing, and say when the costs are equal.

More practice: warm-up

What is a linear system?

What does the solution of a linear system look like on a graph?

Is (0, 0) a solution of y = 2x and y = −x?

What are the intercepts of x + y = 5?

Two lines have the same slope and the same y-intercept. How many solutions does the system have?

More practice: on your own

Solve by graphing: y = 2x + 1 and y = −x − 5.

Solve by graphing: x − y = 2 and x + 2y = 8.

Solve by graphing: y = 3 and x = −2. Describe the lines.

How many solutions does 3x + y = 4 and 6x + 2y = 8 have? Explain.

Two phone plans: Plan A costs $25 a month plus $0.10 per text over the limit; Plan B costs $35 a month with unlimited texts. Write equations and find, by graphing, the number of extra texts where they cost the same.

Challenge

A system's graphs cross at (2.5, −1.5). Why might a student graphing by hand report (2, −2) or (3, −1)? What could they do to be sure?

Write a linear system whose solution is (−1, 4). Use one equation in slope-intercept form and one in standard form, then check.

For which value of b does the system y = 3x + b and y = 3x − 2 have infinitely many solutions? For which values does it have no solution? Can it ever have exactly one solution?

Check your understanding

Quiz

A solution of a linear system in two variables is:

Which ordered pair is the solution of x + y = 5 and x − y = 1?

Which system has no solution?

What are the intercepts of 2x + 5y = 10?

The lines for two snow-plow plans cross at (6, 120). What does this mean?

Why is graphing by hand sometimes not good enough to solve a system?

For parents and teachers

Big idea: a linear system is two equations that must both be true. Its solution is the point where the two lines cross, and there may be one, none or infinitely many solutions. This lesson works toward expectation 1.2 (solving systems by choosing an appropriate graphical or algebraic method) and introduces checking solutions, which is part of expectation 1.1.

How to help: Ask your teen to check every answer by substituting it into both equations; this habit catches most mistakes. Look for real comparisons at home (two phone plans, two rental prices) and ask, "When would they cost the same?"

Common difficulties:

Checking a point in only one equation.
Mixing up the x- and y-intercepts (to find the y-intercept, set x = 0).
Misreading a crossing point that is between grid lines. Encourage graph paper, a ruler, and a check.
Forgetting to answer the real question in words after finding the point (e.g. "Snowbusters is cheaper for more than 6 plows").