--- title: "Solving equations with graphs" subtitle: "Linear relations · Solving and modelling with lines · Lesson 9" --- # Solving equations with graphs **Name:** {{field-24}} **Date:** {{field-15}} **Class:** {{field-15}} > **Key question:** 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. :::teacher-note **In one sentence:** Students solve linear equations by drawing each side as a line and reading the x where the lines meet, then check by substitution (AC9M8A02). **Students should be able to say:** 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. **Width:** the unmarked parts are the floor every student does (predict, run, investigate and make). Typical adds the modify practice; wider adds the real-world contexts. They add contexts, not difficulty. Students work in the online tutorial alongside this sheet. ::: ## Predict $$2x + 3 = 11$$ On the line $y = 11$, every point has $y = 11$. On $y = 2x + 3$, $y$ is 11 at just one value of $x$. :::keep-together **The lines $y = 2x + 3$ and $y = 11$ are drawn. What is $x$ at the point where they meet?** ☐ $x = 3$ ☐ $x = 8$ ☐ $x = 11$ ☐ $x = 4$ My prediction, and why: {{lines-3}} ::: :::teacher-note **What happens:** x = 4. Where they meet, both lines have the same y, so 2x + 3 = 11. That happens when x = 4 (2 × 4 + 3 = 11). The point is (4, 11): 11 is its y, not its x. ::: ## Run Try it on the board in the tutorial. Was your prediction right? What was different? {{lines-2}} ## Reading the solution :::keep-together ### Solving with a graph Here is the method, one line at a time. 1. $2x + 3 = 11$ — Solve this equation with a graph. 2. $y = 2x + 3, \quad y = 11$ — Draw each side as a line. 3. $(4, 11)$ — Find where they meet. There, both sides have the same value, 11. 4. $x = 4$ — The solution is the $x$ of the meeting point, not the $y$. 5. $2(4) + 3 = 11 \; \checkmark$ — Check by substitution in the first equation. **On a graph, $y = 3x - 2$ meets $y = 7$ at $(3, 7)$. What is the solution of $3x - 2 = 7$?** Answer: {{field-15}} ::: :::teacher-note **Answer:** 3 x = 3. Check: 3 × 3 − 2 = 7. ::: :::keep-together ### Find the mistake **Kai solves an equation with a graph. Click the line where his working goes wrong.** ``` 1 Solve x + 5 = 2x + 1 with a graph. 2 Draw y = x + 5 and y = 2x + 1. 3 The lines meet at (4, 9). 4 So the solution is x = 9. ``` Line: {{field-15}} Why is it wrong? {{lines-2}} ::: :::teacher-note **Answer:** line 4 Line 4: 9 is the y of the meeting point, the value of both sides. The solution is the x: x = 4. Check: 4 + 5 = 9 and 2 × 4 + 1 = 9. ::: :::keep-together ### Which equation? **The lines $y = x + 5$ and $y = 2x + 1$ meet at $(4, 9)$. Which equation has the solution $x = 4$?** ☐ $x + 5 = 2x + 1$ ☐ $2x + 1 = 4$ ☐ $x + 5 = 4$ ::: :::teacher-note **Answer:** $x + 5 = 2x + 1$ The meeting point is where the two sides are equal, so it solves x + 5 = 2x + 1. Check: 4 + 5 = 9 and 2 × 4 + 1 = 9. ::: :::typical ## Change the equation **Subtraction.** Each task draws both sides of a new equation. Solve $3x - 4 = 8$. **A negative answer.** Solve $2x + 3 = 1$. **$x$ on both sides.** Solve $x + 5 = 3x + 1$. Now both sides are sloping lines. **Between the lines.** Solve $2x + 1 = 6$. **Never meeting.** $2x + 1 = 2x + 4$. Look at the two lines. What does the last lesson say about them? Work in the tutorial. Plan first: what will you change, and how will you test it? {{box-4}} ::: ## Read the graph :::keep-together ### From a graph **A graph shows $y = 4x - 3$ meeting $y = 9$ at $(3, 9)$. Solve $4x - 3 = 9$.** Answer: {{field-15}} ::: :::teacher-note **Answer:** 3 x = 3. Check: 4 × 3 − 3 = 9. ::: :::keep-together ### Both sides sloping **$y = 2x - 1$ and $y = -x + 5$ meet at $(2, 3)$. What value of $x$ makes $2x - 1 = -x + 5$ true?** Answer: {{field-15}} ::: :::teacher-note **Answer:** 2 x = 2. Check: 2 × 2 − 1 = 3 and −2 + 5 = 3. ::: :::keep-together ### Which equation? **The lines $y = 5$ and $y = x + 2$ meet. Which equation does the meeting point solve?** ☐ $x + 2 = 5$ ☐ $x = 5$ ☐ $x + 5 = 2$ ::: :::teacher-note **Answer:** $x + 2 = 5$ Set the two rules equal: x + 2 = 5. They meet at (3, 5), so x = 3. ::: :::keep-together ### Review **Review: where does $y = 3x - 4$ cross the $y$-axis? Give the value of $y$.** Answer: {{field-15}} ::: :::teacher-note **Answer:** -4 −4: when x = 0, y = 3 × 0 − 4 = −4. ::: ## Make: an equation with a chosen answer Your line is $y = mx + c$. The green point shows where it meets $y = 2x + 1$. Move it so they meet at $x = 3$. (Your line cannot be $y = 2x + 1$ itself.) Stretch: write down your equation, then check $x = 3$ by substitution. Work in the tutorial. Plan first: what will you change, and how will you test it? {{box-4}} ## Exit ticket :::keep-together **The lines y = 2x − 3 and y = 5 meet at (4, 5). Solve 2x − 3 = 5 and check your answer.** {{lines-3}} ::: :::teacher-note **Answer:** x = 4. Check: 2 × 4 − 3 = 5. :::