Graphs
Grade 6-8Graphical Solutions to Equations
Wherever two graphs cross, their equations share a common solution, and reading that crossing point off the graph is often faster than solving algebraically. This lesson covers solving simultaneous equations graphically, solving quadratic equations from where a curve meets a line, setting up the equation needed to find intersection points, and estimating solutions that fall between grid lines.
What you need to know
- The solution to two graphs is where they cross: read off the x and y coordinates.
- To solve f(x) = k, draw the line y = k and find where it meets the curve.
- A curve and a line can cross twice, once, or not at all.
- Rearrange the equation so one side matches a graph you have already drawn.
Written by Asad, co-founder of Teachably
Video walkthrough coming soon
The written lesson below covers everything you need in the meantime.
Solving simultaneous equations graphically
Where two straight lines cross is the one point that satisfies both equations at once. Read the x-coordinate and y-coordinate of the crossing point directly from the graph to give the solution.
The lines y = x + 1 and y = -x + 5 cross at (2, 3), so the solution to the simultaneous equations is x = 2, y = 3.
Solving a quadratic equation graphically
A quadratic curve usually crosses a straight line at two points, giving two solutions. Read the x-coordinate of each crossing point, since these are the values of x that make the equation true.
The curve y = x² - 2x - 3 meets the line y = 5 at x = -2 and x = 4, so these are the two solutions to x² - 2x - 3 = 5.
Setting up the equation for an intersection point
When two graphs are given as equations rather than a picture, set the two expressions for y equal to each other, then rearrange so everything is on one side, giving a quadratic in the form x² + bx + c = 0.
y = x² - 3x + 1 and y = 2x - 5 intersect where x² - 3x + 1 = 2x - 5. Rearranging by moving everything to the left gives x² - 5x + 6 = 0.
Estimating solutions from a graph
When an intersection point does not fall exactly on a grid line, read as precisely as possible between the gridlines to estimate the solution to the level of accuracy asked for, usually 1 decimal place.
The curve y = x² - 2 meets the line y = x + 1 close to x = -1.3 and x = 2.3, read carefully between the grid lines on the graph.
Worked examples
Three exam-style questions, fully solved
The graph shows the lines y = x + 1 and y = -x + 5. Use the graph to solve the simultaneous equations y = x + 1 and y = -x + 5.
Easy- 1.Find the point where the two lines cross on the graph
- 2.Read the x-coordinate of the crossing point: 2
- 3.Read the y-coordinate of the crossing point: 3
Answer: x = 2, y = 3
The graphs of y = x² - 3x + 1 and y = 2x - 5 intersect at two points. Write down the equation that must be solved to find the x-coordinates of these points, giving your answer in the form x² + bx + c = 0.
Medium- 1.Set the two expressions for y equal to each other: x² - 3x + 1 = 2x - 5
- 2.Move all terms to one side: x² - 3x - 2x + 1 + 5 = 0
- 3.Simplify: x² - 5x + 6 = 0
Answer: x² - 5x + 6 = 0
The graph shows y = x² - 2 and the line y = x + 1. Use the graph to estimate the solutions of x² - 2 = x + 1, giving your answers to 1 decimal place.
Hard- 1.Find the two points where the curve and the line cross on the graph
- 2.Read the x-coordinate of the first crossing point as precisely as possible: approximately -1.3
- 3.Read the x-coordinate of the second crossing point as precisely as possible: approximately 2.3
Answer: x ≈ -1.3 or x ≈ 2.3
Practice
6 questions, marked instantly
Type an answer and check it. The worked solution appears once you have had a go. No login needed, and your progress saves in this browser.
Practice
Now try these yourself.
Type your answer and check it. The worked solution appears once you have had a go.
The graphs of y = x + 4 and y = 3x cross at one point. Solve x + 4 = 3x to find the x-coordinate.
The graphs of y = 2x − 1 and y = x + 3 intersect. Work out the x-coordinate of the point where they cross.
Using the previous question, work out the y-coordinate of the intersection.
To solve x² = 2x + 3 using graphs, you draw y = x² and one straight line. Write the equation of that line.
The curve y = x² − 5 and the line y = 4 cross. Solve x² − 5 = 4 to find the two x-values.
The graphs of y = x² and y = x + 6 cross at two points. One is at x = 3. Work out the other x-value.
Avoid these
The mistakes students make most often
Giving only one intersection point for a quadratic equation, when a curve and a line usually cross at two points, giving two solutions.
Reading the y-coordinate instead of the x-coordinate, or mixing the two up, when writing down the solution from where two lines cross.
Making a sign error while moving terms to one side when setting up the equation for an intersection point.
Giving a rounded whole-number answer when the question asks for an estimate to 1 decimal place, instead of reading carefully between the grid lines.
Solving the equation algebraically instead of reading the answer from the graph, when the question specifically asks to use the graph.
FAQ
Questions students and parents ask
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