042i - Solving Linear Equations Graphically

Lesson 42 — Solving Linear Equations Graphically

Strand: Algebra | Descriptor: AC9M8A02 | Duration: 45 minutes

Learning Intentions

• To solve a linear equation by graphing two related lines and reading the point of intersection.
• To connect a graphical solution to its algebraic solution.

Success Criteria

I can:

1. Rewrite a linear equation as two functions, , ready to graph.
2. Plot both lines accurately, by hand and using digital tools, and identify the point of intersection.
3. Read the -coordinate of the intersection as the solution to the original equation.
4. Verify a graphical solution algebraically.

Warmup

(5 minutes — mini whiteboards, rapid recall)

Recall lessons 38–40.

1. State the gradient and -intercept of .
2. Plot two points on without a table.
3. What does the gradient of a line tell you?
4. What does the -intercept tell you?

Answers: 1. Gradient , intercept ; 2. e.g. and ; 3. The rate of change — how much changes per unit increase in ; 4. Where the line crosses the -axis, i.e. the value of when .

Activities

Activity 1 — From Equation to Two Lines (10 min)

I do. Solve graphically.

Step 1 — split into two functions: treat the left side and the right side of the equation as two separate lines to graph.

Step 2 — graph both on the same axes: is a line of gradient and intercept ; is a horizontal line.

Step 3 — read the intersection point: the lines cross at .

Step 4 — state the solution: the -coordinate of the intersection is the solution to the original equation, so .

Step 5 — verify algebraically:

We do: Solve graphically. (Graph and ; intersection at , so .)

You do: Solve graphically, then verify algebraically: (a) (b) .

(Answers: (a) intersection , ; (b) intersection , .)

Activity 2 — Two Sloped Lines: the Variable on both Sides (12 min)

I do. Solve graphically.

Graph and on the same axes. The lines cross at .

Critical point to emphasise: the solution to the equation is only — the -value of is simply where both lines happen to meet, not part of the answer to “solve for .”

Verify algebraically:

We do (digital tool): Using a graphing tool such as Desmos, graph and . Trace to the intersection.

(Answer: intersection , so .)

You do (digital tool or grid paper):

(Answers: 1. intersection , ; 2. intersection , .)

Activity 3 — Applied Inquiry: Comparing Two Plans (12 min)

Pairs, then whole-class share.

Two mobile phone plans are on offer:

• Plan A: 20$0.50$ per gigabyte of data.

• Plan B: 8$1.10$ per gigabyte of data.

1. Write a cost equation for each plan, using for gigabytes and for cost in dollars.
2. Graph both on the same axes (by hand or with a digital tool).
3. At what data usage do the two plans cost the same? Find this graphically, then verify algebraically.
4. For usage above this amount, which plan is cheaper? Justify from the graph.

Socratic scaffolding:

Prompt	Purpose
Understand: what does “cost the same” mean on a graph?	The point where the two lines intersect.
What are the two functions?	and .
Devise a plan	Graph both, or set the expressions equal algebraically.
Carry out the plan	.
Look back — does the graph agree?	The lines should cross at , .
Look back — which plan is cheaper beyond the crossing point?	Compare gradients: Plan B rises faster ( per GB), so beyond it becomes more expensive; Plan A is cheaper for high usage.

Answers: Break-even at GB, costing 302020$ GB, Plan B is cheaper.

Checks for Understanding

(6 minutes — exit ticket, collected)

1. Write the two functions you would graph to solve .
2. Two lines, and , intersect at . What equation does this solve, and what is the solution?
3. Solve graphically or algebraically, and state which method you used and why.
4. Reasoning. Explain why the -coordinate of the intersection point is not the answer when solving a one-variable equation graphically.

Answers: 1. and ; 2. , solution ; 3. (method choice justified either way); 4. The equation asks only for the value of that makes both sides equal; the -value is simply the common output value at that point, not part of what was asked.

Common Misconceptions

Misconception	How to pre-empt it
Reading off the -coordinate of the intersection as the answer.	Always underline “solve for ” and circle only the -coordinate.
Graphing only one side of the equation.	Insist both sides are written as separate functions before any plotting begins.
Assuming lines that look close together intersect where they appear to cross on a rough sketch.	Always verify graphical answers algebraically, as in every worked example here.
Believing a horizontal line like has no “equation” worth graphing.	Show explicitly that is a valid, flat line — treat it identically to a sloped one.
Confusing gradient sign with which line is “winning” at large .	Use the phone-plan task: the line with the smaller gradient becomes cheaper for large , even though it may start higher.

Enrichment — Competition-Style Problems

E1 (Kangaroo style). The lines and intersect at a point. Find the sum of the coordinates of the intersection point.

Answer

Sum .

E2 (AMC Junior style). Two lines with the same gradient are graphed to “solve” an equation. What happens, and what does this tell you about the original equation?

Answer

Parallel lines with the same gradient never intersect (unless they are identical). If the intercepts differ, the equation has no solution; if the lines coincide, every value of is a solution.

E3 (Challenge). Without graphing, predict the intersection point of and , then verify.

Answer

Both share intercept , so they cross the -axis at the same point: . Check: and ✓

E4 (Challenge). Three lines , and are graphed together. Find the two points where meets the other two lines, and determine whether all three lines pass through a common point.

Answer

and : , point .

and : , point .

Both intersections are the same point, so all three lines are concurrent at .

Homework

1. Write the two functions you would graph to solve each equation: (a) (b) (c) .
2. Solve each equation from Q1 graphically (sketch or digital tool) and verify algebraically.
3. Two lines and intersect at . Write the equation this solves and state the solution.
4. A taxi company charges 4$2$1$ per kilometre with no flat fee. (a) Write a cost equation for each. (b) Find graphically where the costs are equal. (c) Verify algebraically.
5. Reasoning. Explain, using a sketch or description, what it means graphically when a linear equation has no solution.
6. Challenge. The lines and intersect at . Find .

Answers: Q1 — (a) , (b) , (c) , . Q2 — (a) (b) (c) . Q3 — , . Q4 — (a) , (b) intersection at (not physically meaningful — discuss) (c) ; since distance can’t be negative, the rideshare is cheaper for all realistic distances. Q5 — the two lines are parallel (equal gradients) and do not coincide, so they never cross; there is no intersection point and hence no solution. Q6 — at , ; so , giving .