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Using Graphs to Solve Equations

Using and creating graphs to solve quadratics and pairs of simultaneous equations.
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Equations:

\(2y = x + 5\)

\(2y+x+5=0\)

\(y=(x+1)^2\)

\(y=x^3 -11x^2 +39x -45\)

\(y=-x^3 -11x^2 -39x -49\)

\(x^2 + y^2 = 25\)

There are six equations shown to the left and their graphs are shown above. Your first job is to work out which equation describes which graph.

Use the graphs to find a solution to the following simultaneous equations:

\( \quad \quad 2y = x + 5\)
\( \quad \quad 2y + x + 5 = 0\)

\(x =\) ☐ ✓ ✗ ☐\(y =\) ☐ ✓ ✗

3

Use the graphs to find a positive value of \(x\) which satisfies the simultaneous equations:

\( \quad \quad 2y = x + 5\)
\( \quad \quad x^2 + y^2 = 25\)

☐\(x =\) ☐ ✓ ✗

4

Use the graphs to find a negative value of \(y\) which satisfies the simultaneous equations:

\( \quad \quad 2y + x + 5 = 0\)
\( \quad \quad x^2 + y^2 = 25\)

☐\(y =\) ☐ ✓ ✗

5

Use the graphs to find a negative value of \(x\) which satisfies the simultaneous equations:

\( \quad \quad y = (x + 1)^2\)
\( \quad \quad x^2 + y^2 = 25\)

☐\(x =\) ☐ ✓ ✗

6

Use the graphs to find an integer (whole number) value of \(x\) which satisfies the simultaneous equations:

\( \quad \quad y = x^3 -11x^2 +39x -45\)
\( \quad \quad x^2 + y^2 = 25\)

☐\(x =\) ☐ ✓ ✗

7

Find the value of \(x\) which satisfies the simultaneous equations:

\( \quad \quad y = -7\)
\( \quad \quad y=-x^3 -11x^2 -39x -49\)

☐\(x =\) ☐ ✓ ✗

8

Find the value of \(y\) which satisfies the simultaneous equations:

\( \quad \quad x = 2\)
\( \quad \quad y = x^3 -11x^2 +39x -45\)

☐\(y =\) ☐ ✓ ✗

9

Use the graphs to find a value of the largest solution of the equation:

\( \quad \quad (x + 1)^2 - 3 = 1\)

☐\(x =\) ☐ ✓ ✗

10

Use the graphs to find a value, to one decimal place, of the smallest solution of the equation:

\( \quad \quad (x + 1)^2 - \frac{x}{2} - \frac{5}{2} = 0\)

☐\(x =\) ☐ ✓ ✗

Check

This is Using Graphs to Solve Equations level 2. You can also try:
Level 1 Level 3 Level 4

Instructions

Try your best to answer the questions above. Type your answers into the boxes provided leaving no spaces. As you work through the exercise regularly click the "check" button. If you have any wrong answers, do your best to do corrections but if there is anything you don't understand, please ask your teacher for help.

When you have got all of the questions correct you may want to print out this page and paste it into your exercise book. If you keep your work in an ePortfolio you could take a screen shot of your answers and paste that into your Maths file.
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Mathematicians are not the people who find Maths easy; they are the people who enjoy how mystifying, puzzling and hard it is. Are you a mathematician?

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Description of Levels

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Level 1 - The original basic exercise

Level 2 - A collection of graphs already drawn with related questions

Level 3 - Graphs to draw with pencil and paper in order to find solitions

Level 4 - For older students who have access to a Graphic Display Calculator (GDC).

Exam Style Questions - A collection of problems in the style of GCSE or IB/A-level exam paper questions (worked solutions are available for Transum subscribers).

More Graphs including lesson Starters, visual aids, investigations and self-marking exercises.

More Simultaneous Equations including lesson Starters, visual aids, investigations and self-marking exercises.

Answers to this exercise are available lower down this page when you are logged in to your Transum account. If you don’t yet have a Transum subscription one can be very quickly set up if you are a teacher, tutor or parent.

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Help Video

Don't wait until you have finished the exercise before you click on the 'Check' button. Click it often as you work through the questions to see if you are answering them correctly. You can double-click the 'Check' button to make it float at the bottom of your screen.

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