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G11 · Find a linear-system intersection graphically
Learn to find a linear-system intersection graphically through clear examples and targeted practice.
Ontario Grade 10 Mathematics
Modelling Linear Relations
Use a graph to find the solution to a linear system
A community centre charges a registration fee and an amount for each class. Another centre may use a different fee plan. A graph can show when the two plans cost the same. The point where the graphs meet gives that information. In this lesson, you will use a graph to find that meeting point. You will not need to find it by rearranging or solving equations.
What you will learn
- Recognize a linear system as two linear relations shown together.
- Read the point where two graphed lines intersect.
- State the graphical solution as an ordered pair and explain what it means.
- Recognize when a graph shows no intersection or the same line.
1. Bridge from points and lines
A coordinate grid has a horizontal axis and a vertical axis. A point on the grid is named by an ordered pair: the first number tells how far to move horizontally, and the second tells how far to move vertically. For example, is 2 units right and 5 units up from the origin, where the axes cross.
A linear relation makes a straight line when graphed. One way to draw a line is to plot points that belong to it and connect them with a straight edge. A linear system is a set of two linear relations considered together. Its graph contains both lines on the same coordinate grid.
Before reading an intersection, check that both lines use the same axes and scale. A scale is the amount represented by each grid interval. If one interval means 1 unit on the horizontal axis, it should be clear what the vertical intervals mean too.
- An ordered pair names a point on a grid.
- A linear relation graphs as a straight line.
- A linear system has two relations, so its graph has two lines.
2. What an intersection tells you
An intersection is a point where two lines cross. At that point, the same horizontal and vertical coordinates belong to both lines. So the intersection is a shared solution: it makes both relations true at the same time.
For example, suppose one line describes the cost at Centre A and another describes the cost at Centre B. The horizontal coordinate could represent the number of classes, and the vertical coordinate could represent the cost in dollars. If the lines meet at , then at 4 classes both plans cost CAD 30. The meaning depends on the labels and units on the graph.
A graphical solution is the intersection point read from a graph. Read across the horizontal axis first, then up or down to read the vertical coordinate. Give the result as an ordered pair, and include units or context when they are known. If the crossing falls between grid marks, estimate carefully and state that the value is approximate.
Two different lines might not cross in the part of the graph shown. If they are parallel, they do not meet, so there is no shared solution. If the lines lie exactly on top of each other, every point on that line is shared. On a graph, trace both lines to check whether they are separate or coincident; coincident means they occupy the same positions.
- The intersection satisfies both relations.
- The horizontal coordinate is read first.
- A single crossing gives one graphical solution; no crossing gives none; one shared line gives many.
3. Read the graph carefully
A clear graph makes an answer easier to trust. Give each axis a label, choose a scale that fits the useful points, and plot points accurately. Use a ruler or a graphing tool to draw each straight line. A graphing tool can also display lines from given equations. In either case, the goal here is to interpret the graph, not to use an algebraic solving method.
When a graph is supplied, first identify which line represents each relation. Then locate where the two lines meet. Follow the grid lines or tick marks to read the horizontal coordinate and the vertical coordinate. Check the scale on each axis separately. Do not assume that one grid square represents the same amount on both axes.
A graph may only show part of a line. If the lines appear not to meet in the visible window, do not immediately conclude that there is no solution. The crossing could be outside the displayed region. Read the question and graph range carefully. A conclusion of no intersection is justified only when the graph or the information provided establishes that the lines do not meet.
- Check the labels and scale before reading coordinates.
- Trace each line to its crossing, rather than choosing a nearby point.
- A limited viewing window may hide an intersection.
4. Guided example: compare two plans
A graph shows two straight-line cost plans. The horizontal axis is the number of classes, and the vertical axis is the total cost in dollars. The graph was made from these relations: Centre A charges a starting fee of CAD 6 and CAD 6 per class; Centre B charges a starting fee of CAD 18 and CAD 3 per class. Find the intersection graphically and explain its meaning.
To make a hand-drawn graph, plot several convenient points for each line. For Centre A, the points include , , and . For Centre B, they include , , and . Plot and connect each set with a straight line on the same grid. These points are graphing aids; the answer comes from reading where the lines meet.
At 4 classes, both lines pass through the same point on the graph. The vertical coordinate there is 30. The lines therefore intersect at . This point is a solution to the system because it belongs to both lines. In context, the plans cost the same after 4 classes, and that cost is CAD 30.
- Graph both relations on the same coordinate grid.
- Read the intersection as horizontal coordinate first, then vertical coordinate.
- Explain the coordinates using the graph’s labels and units.
Reading a graphical solution
| What the graph shows | Graphical result | Meaning |
|---|---|---|
| Two separate lines cross once | One intersection point | One shared ordered pair |
| Lines do not meet | No intersection | No shared solution |
| Both relations are the same line | The lines overlap | Every point on that line is shared |
Worked example
Find the plans’ break-even point from a graph
Use the graphing information in Section 4 to find where Centre A and Centre B have the same cost. State the solution and its meaning.
- Identify the axesThe horizontal coordinate represents the number of classes, and the vertical coordinate represents total cost in dollars. This tells us what each coordinate will mean.
- Locate the crossingPlot the listed points for each plan and draw both lines. The lines meet where both include the point at 4 classes and a cost of 30 dollars.
- State the graphical solutionWrite the shared point as an ordered pair, with the number of classes first. In context, both plans charge CAD 30 for 4 classes.
Answer: The lines intersect at . The plans cost the same after 4 classes: CAD 30.
Check: At 4 classes, Centre A’s listed cost is CAD 30 and Centre B’s listed cost is also CAD 30, so the point belongs to both lines.
Common mistakes and how to avoid them
Writing the coordinates in the wrong order.
Correction: Read the horizontal coordinate first and the vertical coordinate second. The point is written as .
Reporting a point that lies on only one line.
Correction: Check that the selected point is where both lines meet, not just a point on one of them.
Ignoring the graph’s scale or labels.
Correction: Read the tick marks and axis labels before assigning values to the intersection coordinates.
Calling lines non-intersecting because they do not cross in the visible window.
Correction: Check whether the graph is cut off. A crossing may lie beyond the displayed range.
Lesson summary
- A linear system graphs as two straight lines on one coordinate grid.
- The point where the lines meet is their shared solution.
- Read the horizontal coordinate first and the vertical coordinate second.
- Use axis labels to explain what the ordered pair means.
- Check whether the lines cross once, never meet, or overlap.
Check your understanding
Question 1
On a graph, the horizontal axis is time in minutes and the vertical axis is distance in kilometres. Two lines cross at . What does the intersection mean?
- At 6 minutes, both distances are 15 kilometres.
- At 15 minutes, both distances are 6 kilometres.
- One line has a distance of 6 kilometres and the other has 15 kilometres at the crossing.
- The lines never have the same distance.
Show answer and explanation
At 6 minutes, both distances are 15 kilometres.
The first coordinate is read from the horizontal axis, so it is 6 minutes. The second is read from the vertical axis, so both relations show 15 kilometres then.
Question 2
Two separate straight lines on a graph cross at exactly one point. How many shared solutions does the system have?
- None
- One
- Two
- Every point on either line
Show answer and explanation
One
The single crossing is the only point that belongs to both lines, so it gives one shared solution.
Key terms
- Coordinate grid
- A grid with horizontal and vertical axes used to locate points.
- Ordered pair
- Two coordinates written in order, horizontal first and vertical second.
- Linear relation
- A relation whose graph is a straight line.
- Linear system
- Two linear relations considered together.
- Intersection
- A point where two lines meet.
- Graphical solution
- The point read from a graph that belongs to both lines.
Continue through MFM2P
View the complete MFM2P Ontario Grade 10 Mathematics curriculum and lessons
- G1 · Find where two linear models have the same value
- G2 · Solve first-degree equations including fractional coefficients
- G3 · Isolate and evaluate a variable in a formula
- G4 · Convert a line equation to slope-intercept form
- G5 · Connect rate of change to slope as rise over run
- G6 · Identify slope-intercept form and horizontal or vertical lines
About this lesson
Published by DoAssignment. This AI-assisted lesson follows Ontario Grade 10 Mathematics (MFM2P), study topic G11. It is a study resource, not an official curriculum publication.