# Graphical Solutions to Mathematical Problems

Students are required to use linear and quadratic graphs to estimate values of y for given values of x and vice versa. The most common area of use for this skill is when students are required to solve equations graphically. Students are also expected to be able to find approximate solutions of simultaneous equations using a graphical method.

Visit the secondary mathematics webpage to access all lists.

### Graphs

This resource contains a variety of activities requiring students to draw linear and quadratic graphs. Students need to appreciate that graphs can then be used to solve equations and need to be able to draw appropriate graphs in order to complete this task.

In pack one of this resource, the activity **Solving by graphs** requires students to complete a mapping, produce a set of coordinates and then draw the graph. The next step is then to use the graph produced to solve an equation. Students then use the mapping to check the solution.

The activity **Simultaneous equations **from graphs suggests a good way to show how to solve simultaneous equations by drawing graphs on transparencies and overlaying one graph on top of the other.

**Calculator graphs **gives suggestions as to how graphical calculators can be used to solve equations using graphs.

In pack two, **Quadratic solutions **is an activity requiring students to solve quadratic equations graphically.

### Linear Graphs and Equations

The first part of the text recaps how to draw linear graphs. The section appropriate to this topic is section 5.3, beginning on page 15 of the pdf. The section begins with a recap of how to solve linear equations algebraically. Explanations, examples and exercises follow, in which the process of solving equations using a graph is explored. Examples include two graphs being drawn in order to solve linear equations with an unknown on both sides.

Section 5.5, beginning on page 28 of the pdf, contains explanations, examples and exercises exploring how graphs can be used to solve simultaneous equations.

### Equations

The simultaneous equations resource contains a number of suggestions of tasks, questions and puzzles which can be used as introductory tasks, extension tasks or tasks to assess understanding. The problems can be posed requiring students to form suitable equations and draw appropriate graphs which can be used to solve the problem. Students should be encouraged to explain their work at each stage and to show how they have checked their solution.

### Foundation 2

Chapter seven of this text, **Using graphs to solve problems**, explores a range of aspects of the uses of graphs in a variety of contexts. **Using graphs to solve equations**, beginning on page 142 of the pdf, requires students to be able to plot linear and quadratic graphs and then use these graphs to answer questions. Students are asked to find where graphs cross axes, find approximations for square roots, solve linear equations and read values from graphs.

### Number and Algebra 4

This text book contains three units which contain useful ideas and resources when planning this topic.

Tasks in Unit 4: **Linear graphs**, beginning on page 37 of the pdf, require students to draw linear graphs using suitable technology and to read off where the graphs cross the axes, use the graphs to solve equations. In section C: **Using linear graphs**, students are required to use linear graphs to solve problems in a variety of contexts.

The activities in Unit 6: **Simultaneous equations**, beginning on page 57 of the pdf, looks at problems which can be solved by forming and solving simultaneous equations. Section C: **Using graphs **explains how graphs can be used to solve the problems outlined in the first section.

The content of Unit 9:** Graphs **covers the drawing and use of non-linear graphs. Section B: **Graphs from equations**, beginning on page 90 of the pdf, requires students to plot the graphs of quadratic equations and use the graphs to solve quadratic equations.

### Graphical Solution of Equations

This Virtual Textbook resource is an interactive excel program to enable students to investigate the solution of polynomial equations up to cubics using a graphical method.

The program shows the graphs of cubic equations and the equivalent tables of values. There is a set of questions requiring students to solve equations graphically and confirm their solutions algebraically. They can amend the coefficients of cubic, quadratic or linear equations in order to draw the graphs and see the points of intersection.

One interactive sheet shows the graph of a cubic function and a straight line. Students alter the gradient and intercept of the straight line in order to solve an equation related to the given cubic function.

There are five sheets of questions suitable for duplication and use in the classroom. These are mainly related to simultaneous linear equations and problems related to quadratic equations.