Solving equations with graphs
Draw each side of an equation as a line. Where the lines meet, both sides are equal, so the x there is the solution.
A graph can solve an equation you have no method for yet: draw both sides and look where they meet. Calculators, spreadsheets and engineers do exactly this.
An equation such as says two things are equal. Each side can be drawn as a line: , and (a flat line, because 11 does not change with ). In the Equations unit you solved equations with a balance and by backtracking. Now you solve them by looking at a graph.
- Predict: Read the problem and decide what you think before you try anything. A wrong guess is fine: it shows you what to look for.
- Try: Try it on the board and compare the result with your prediction.
- Investigate: Work out why it happens, one step at a time.
- Modify: Change the problem on purpose and see what happens.
- Practise: A few on your own, to check you have it.
- Make: Make your own problem using what you have learned. Expect a few tries; that is normal.
Make your prediction first. Help opens once you've locked it in.
Where they meet
On the line , every point has . On , is 11 at just one value of .
Find the meeting point
Drag until it is on both lines. Read its coordinates under the board. Which number is the solution of ?
Drag to the point where the two lines meet.
Use the board. This ticks when you get there.
Reading the solution
Here is the method, one line at a time.
- Solve this equation with a graph.
- Draw each side as a line.
- Find where they meet. There, both sides have the same value, 11.
- The solution is the of the meeting point, not the .
- Check by substitution in the first equation.
Read Kai's working one line at a time. Which number did he take from the point?
The meeting point solves the equation that sets the two rules equal.
On a graph, meets at . What is the solution of ?
Kai solves an equation with a graph. Click the line where his working goes wrong.
Change the equation
Each task draws both sides of a new equation. Solve .
Solve .
Need a hint?
The lines meet to the left of the y-axis.
Solve . Now both sides are sloping lines.
Need a hint?
Both lines slope now. Follow them until they cross.
Solve .
Need a hint?
P moves in halves on this board. The lines meet between two grid lines.
. Look at the two lines. What does the last lesson say about them?
Solve : drag to where the lines meet.
Use the board. This ticks when you get there.
Solve : drag to where the lines meet.
Use the board. This ticks when you get there.
Solve : drag to where the lines meet.
Use the board. This ticks when you get there.
Solve : drag to where the lines meet.
Use the board. This ticks when you get there.
Read the graph
A graph shows meeting at . Solve .
and meet at . What value of makes true?
Review: where does cross the -axis? Give the value of .
An equation with a chosen answer
Your line is . The green point shows where it meets . Move it so they meet at . (Your line cannot be itself.)
Stretch: write down your equation, then check by substitution.
Need a hint?
At x = 3, the line y = 2x + 1 is at 7. Your line must pass through (3, 7).
Another hint
Pick m first. Then find c so that 3m + c = 7.
Set and so that the equation has the solution .
Use the board. This ticks when you get there.
For teachers
Students solve linear equations by drawing each side as a line and reading the x where the lines meet, then check by substitution (AC9M8A02). About 45 minutes.
Open the teacher guide: answers, solutions, slides and a worksheet.
Students will be able to
- I can solve an equation by drawing each side as a line and finding where they meet.
- I can explain why the solution is the x of the meeting point, not the y.
- I can check a graph's answer by substitution, and recognise parallel lines as no solution.
Before class
Do the Run and Modify boards yourself first. This lesson joins the two units: the Equations unit's equations (solved with the balance and backtracking) and this unit's lines.
The key idea is which number to read. At the meeting point both sides have the same value (the y), and the solution is the x that gives it. Expect students to read the y: it is the most common mistake, and the Predict and Kai's working are built around it.
The Make board uses two sliders and shows the meeting point in green. Any line through (3, 7) other than y = 2x + 1 works.
Running the lesson
- Predict: take a show of hands. 'x = 11' is the common wrong answer (reading the y). Do not correct it: the Run board settles it.
- Run: when P is on both lines, ask: what is the same about the two lines at this point? (The y.) So which number is the solution? (The x.)
- Investigate: model the worked example live. Sketch the two lines, mark where they meet, then say 'the solution is the x, because that is the input that makes both sides equal'. Finish with the substitution check.
- Modify: the last task has parallel lines. Ask: what did the last lesson say about lines with the same m? Then: what does that mean for the equation?
- Practise: for each answer, ask a student to say which number is the y and why it is not the answer.
- Make: collect two different lines that work and compare them: both pass through (3, 7).
When students get stuck
- A student gives the y of the meeting point as the solution. Ask: Put your answer into the equation. Is it true? Fix: At the meeting point, read the x off the bottom axis. The y is the value of both sides there, not the answer.
- A student cannot find where the lines meet on the negative task. Ask: Which way do the lines get closer together? Fix: Follow both lines to the left of the y-axis. The solution can be negative.
- A student says parallel lines meet somewhere far away. Ask: How far apart are they at x = 0? At x = 10? Fix: Work out both rules at a few values of x: the gap is always 3. They never meet, so there is no solution.
- A student moves the Make sliders at random. Ask: Where must your line go through? Fix: Work out y = 2x + 1 at x = 3 (it is 7). Choose m, then find c so that 3m + c = 7.
Stretch and support
Stretch: Solve 3x − 2 = x + 5 with a graph: the answer is not a whole number. How accurate can a graph be? (x = 3.5)
Support: Give squared paper and have students draw both lines from tables of values (x = 0 to 4) before using the board.
Where it leads
Next: modelling with two lines. Two phone plans or two ways to pay become two rules; where their lines cross is where they cost the same.
Where it leads in class: break-even points in business, when two runners meet, and solving equations that have no neat algebra method.
What each step is for
- where they meet: Committing to which number the meeting point gives (x or y) is the whole lesson: students who read y here will read y on every graph.
- find the meeting point: Dragging P onto both lines at once makes 'where they meet' physical: there is exactly one such point, and its x makes both sides equal.
- reading the solution: The solution is the x of the meeting point, never its y; and a graph's answer is still checked by substitution.
- change the equation: One change at a time: a subtraction, a negative solution, x on both sides, an answer between the grid lines, and two lines that never meet (no solution).
- read the graph: Reading the solution as the x of a meeting point, and naming the equation it solves, in mixed forms; the review keeps c fresh.
- an equation with a chosen answer: Building a second line through a chosen meeting point reverses the lesson: from the solution to the equation.
- Solve an equation by drawing each side as a line and finding where they meet.
- Explain why the solution is the x of the meeting point, not the y.
- Check a graph's answer by substitution, and recognise parallel lines as no solution.