Stage 4 - algebra – solve linear equations using graphical techniques

Strategy

Students can:

  • Recognise that each point on the graph of a linear relationship represents a solution to a linear equation
  • Graph two intersecting lines on the same set of axes and read off the point of intersection
  • Understand that point of intersection of two lines represents the only solution that satisfies both equations
  • Solve linear simultaneous equations by finding the point of intersection of their graphs

Activities to support the strategy

There are three strategies that students may use to find the point of intersection of two linear equations.

The first strategy is for students to carefully graph the equations on a number plane. An accurate graph will allow the point of intersection to be read off.

The second strategy relies on students understanding that each point on a line represents the coordinate values which satisfy the linear equation. In this way the point of intersection can be found by substituting to find values which satisfy bath linear equations.

The third strategy is to solve the linear equations simultaneously. This algebraic approach will yield the point of intersection.

Activity 1

Students should review the process of graphing straight lines. BBC Bitesize graphs resource allows students to review and practice the process of graphing straight lines.

Students should then practice graphing pairs of linear equations. Reading off the point of intersection will yield the answer required. Students are encouraged to view the video on the Solving linear simultaneous equations graphically activity, modelling the process.

Activity 2

Use a graphing software program such as GeoGebra may be used to graph pairs of linear equations and find their points of intersection. A number of online examples and tutorials are also available.

Activity 3

More advanced students should investigate the algebraic process of solving simultaneous equations. There are two main methods, elimination and substitution. Both methods should be modelled and students given adequate practice to build confidence.

A number of online resources may be used for this purpose.

References

Australian curriculum reference: ACMNA194

Solve linear equations using graphical techniques

ACMNA237: Solve linear simultaneous equations, using algebraic and graphical techniques, including with the use of digital technologies

NSW syllabus reference: MA4-11NA

Creates and displays number patterns; graphs and analyses linear relationships; and performs transformations on the Cartesian plane

MA5.2-8NA; Solves linear and simple quadratic equations, linear inequalities and linear simultaneous equations, using analytical and graphical techniques

Student resources

This interactive resource introduces students to the equations of vertical and horizontal lines. Students are involved in finding the coordinates of points on a straight line whose equation is known and finding the rule from a graph. This resource is one of a series of online resources from the BBC's Bitesize collection.

These illustrated information sheets review inequality symbols, the representation of inequalities on a number line and the solution of inequalities with a positive coefficient of the unknown. Simultaneous equations are defined and simple pairs of equations are solved using substitution, algebra and graphical representation. Students solve inequalities and simultaneous equations with feedback obtained by selecting the Show solutions tab.

Linear simultaneous equations can be solved graphically. All you need to do is graph each equation and find the x and y coordinates where they intersect. Then just write these as values for x and y. Watch how it's done. You'll be an expert in no time!

An open source dynamic mathematics software for learning and teaching at all levels. A geometry package providing for both graphical and algebraic input.

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