Linear expressions and equations: Year 10: Planning tool

Expected level of development

Australian Curriculum Mathematics V9: AC9M10A02

Numeracy Progression: Number patterns and algebraic thinking: P9

At this level, students have developed confidence from their prior learning to algebraically solve linear equations of the form ax + b = c and can represent the solution graphically. They extend this to solving pairs of simultaneous linear equations with two variables and representing their solutions graphically. Students also graph and obtain the point of intersection of the graphs of two linear functions of the form ax + by = c.

They will have previously learnt to solve linear inequalities in one variable and represent their solution on a number line. They extend this to graphing regions corresponding to linear inequalities of the form ax + by < c and ax + by > c and identify whether a point belongs to the solution region in the Cartesian plane.

They are able to discuss how the linear equation ax + by = c represents the boundary of the two regions and is not included when the inequality is expressed using < or >, or is included when the inequality is expressed using ≤ or ≥.

Present opportunities for students to solve problems involving linear inequalities and simultaneous linear equations in two variables in real-world contexts. This involves the processes of formulation, solution and interpretation, and communication of results in context. Financial contexts are often used for this sort of problem and can be used to introduce simple cases with decimal coefficients.

Teaching and learning summary:

Revise solving linear equations and inequalities in one variable, and their graphical representation.
Introduce linear functions expressed in the form ax + by = c and graph these functions.
Graph linear inequalities in two variables, identify the corresponding regions on the Cartesian plane, and identify which region a given point is in, or whether it is on the boundary line.
Demonstrate how to solve simultaneous linear equations algebraically using substitution and elimination methods, and how to represent and interpret the solution graphically.
Discuss how to identify and graph parallel lines for linear functions of this form.
Use problems that involve real-world contexts and data so that students understand how these more complicated forms can be used to solve a broader range of practical problems.

Students:

graph a linear function of the form ax + by = c and calculate its intercepts and gradient
solve a pair of simultaneous linear equations in two variables and represent the solution graphically as a point of intersection on the Cartesian plane
identify the two regions formed on the Cartesian plane by a linear function of the form ax + by = c and test whether a given point belongs to a region or is on the boundary of the region
use information from real-world contexts to formulate and solve related problems using linear functions and inequalities
interpret and communicate solutions involving linear function or inequalities with respect to a practical context.

Some students may:

not have consolidated the meaning of the inequality symbols and how to interpret them appropriately in context.
have difficulty understanding ax + by = c, where b is non-zero, as an alternative form of rule for a linear function.
misinterpret what the intersection of two lines represents on the Cartesian plane.
not understand the role of a test point to determine a region on the Cartesian plane.

The Learning from home activities are designed to be used flexibly by teachers, parents and carers, as well as the students themselves. They can be used in a number of ways including to consolidate and extend learning done at school or for home schooling.

Learning intention

We are learning to graph linear functions with rules of the form ax + by = c, and identify the regions corresponding to the inequalities ax + by < c and ax + by > c.
We are learning to solve a pair of simultaneous linear equations with rules of the form ax + by = c algebraically, and to represent the solution graphically as the point of intersection of two straight lines.

Why are we learning about this?

Finding where the graphs of two linear functions intersect enables us to solve practical problems such as choosing when to switch from one supplier to another and optimising outcomes such as cost for a task. Linear inequalities enable us to decide which values meet acceptable conditions for solving a problem, for example, making sure that the combined cost of two items does not exceed a budgeted maximum amount.

What to do

Practise graphing linear functions and related regions on the Cartesian plane

Question 1

Draw the graph of the linear function 2x + 3y = 24 by hand on graph paper.
Find the coordinates of the two axis intercepts.
Shade the regions corresponding to points where 2x + 3y > 24 and points where 2x + 3y < 24 in two distinct colours on the graph.
Which region is the point (0, 0) in? Give the coordinates of a point that lies in the other region.

Question 2

Check your answers from question 1 using either a graphics calculator or an online program, such as GeoGebra.
Graph 2x + 3y = 24, 2x + 3y > 24 and 2x + 3y < 24 on the same set of axes and check that your points lie in different regions.
Repeat for the linear functions with rules:

2x – 3y = 24 5x + 4y = 40 –3x + 9y = 36

Practise solving pairs of simultaneous linear equations algebraically and representing their solution graphically

Question 3

Draw the graphs of 2x – y = 6 and x + 2y = 8 by hand on the same graph.
State the coordinates of their point of intersection. Use either a graphics calculator or GeoGebra to check your graphs.
Now solve these rules algebraically as a pair of simultaneous linear equations.
Verify your answer by substituting the values for x and y back into each rule.
Repeat the steps for the pairs of simultaneous linear equations below; then write down rules for a pair of linear functions of the form ax + by = c whose graphs do not intersect.
5x + 3y = 48 and 4x – 2y = 12 5x – 3y = 48 and 2x – 6y = 12
Show this on a graph. What mathematical property of the graphs of linear functions does this illustrate?

Success criteria

I can draw the graph of a linear function with a rule of the form ax + by = c, and identify the inequalities corresponding to the regions above and below the graph on the Cartesian plane.
I can solve a pair of simultaneous linear equations with rules of the form ax + by = c algebraically and represent the solution graphically.

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