1.4 Linear Algebra

(AS 91029 - 3 credits)

Click here to access the recommended Learning Plan for this topic

Linear Algebra is the process of modelling a consistently increasing relationship using an algebraic expression and then using that model to solve problems.

Linear Algebra Skills Student Unit Plan - with links

LINEAR ALGEBRA REVIEW CHOICE BOARD

Using and solving linear models including simultaneous equations.pdf

Methods you will need to demonstrate for success in this standard:

You will find separate sections about each of these skills a little further down this webpage.

Read values from a graphs or a table to answer a problem
Use a linear equation to answer a problem
Forming a linear model from a table, graph or words by
- Finding the gradient of a linear model
- Finding the y-intercept of a linear model
Graph a linear model from an equation
Find the Value of a Line at a Specified Point or filling a table using an Equation (Substitution)
Find an intersection point between two lines to answer a problem

Desmos

Desmos is a very useful tool for this topic. It will help you to visualise the equations and relationships you come across. Use the links here to explore and use this resource.

Tutorial on how to use DESMOS to make a linear graph

Changing graph settings

Teacher Desmos Activities - your teacher may set some of these for you in your google classroom to do with your class, but if not, here are some activities that help you to see how linear algebra and graphs work:

Linear Graphs Turtle Races Land the Plane Marbleslides

#1 Read Values from a Graph or a Table to Answer a Problem

To do this, you must be able to look at a graph and read where a point is in relation to the x and y axis. For example, on the graph to the right, the point on the line is at 2, -1.2.

In order to do this, you just have to look at where the point is and draw a line up and a line across and find where those two lines intersect the x and y axes.

To read from a table, you must be able to look at a table and read where a point is in relation to the x and y axis. For example, on the table to the right, the point at 2 is -1.2.

In order to do this, you just have to look at the table and look for the number you know and read the corresponding value on the table.

#2 Use a Linear Equation to Answer a Problem

To do this, you must be able to use a given or derived linear equation (y=mx+c) to find the answer to a problem.

For example, you could be told that income (I), in terms of time (T), is I=2T-3 and asked what your income would be after 3 time periods.

For this, you'd put 3 in place of T, giving you:

I=(2*3)-3=6-3=3

Therefore your income would be 3 after 3 time periods.

#3 Forming a Linear Model from a Table, Graph or Words

Gradients

To do this, you must be able to find how much a pattern goes up for every unit it goes across.

For example, you are given the following situation:

Martha works 8 hours a day and earns $120 after tax. How much is her after tax hourly rate?

To do this, you figure out which is the y variable and which is the x variable (Hint: Time is almost always the x variable). Here, hours is the x variable and money is the y variable. You then divide the y by the x, like so:

$120/8 hours=$15/hour

As such, you can say that her after tax hourly pay is $15 per hour.

y-intercepts

To do this, you must find the point at which the x variable is equal to 0.

For example, if we are given the equation y=2x+5, and we substitute in 0 for x, we get this:

y=(2*0)+5=0+5=5

Therefore the y-intercept for this is 5.

Putting the gradient and intercept together

This online explorer, will help you to see how m (gradient) and b (y-intercept) change the graph y=mx+b. Play around with it and then have a go at the activity on the link below it.

Videos to support your learning of Slopes, and how we put the y-intercept together with the gradient to find the equation.

Back to basics on 'slope'

Graphing y=mx+b

Finding equation from a table of values (harder examples)

A Game to check your understanding

function machine game

finding a rule for a linear pattern,

finding a rule from a pattern in context (very detailed but long)

Finding equation from a table of values (harder examples)

Finding a rule for a linear pattern:

nth_Term.ppt

Linking the Equation>Table>Graph matching activity in context

#4 Graph a Linear Model from an Equation

To do this, you need to be able to plot point on a graph based on what an equation tells you.

For example, if you are told to graph the equation y=7x-2, you would do it by finding two points on the graph. This can be done by picking any two values for x and substituting it in. Easiest option is to pick x=0 and x=1, as shown here:

y=(7*0)-2=-2

y=(7*1)-2=5

Therefore two points on your graph are (0,-2) and (1,5). Join up the two points and continue the line in both directions AS LONG AS THE RELATIONSHIP IS CONTINUOUS!

#5 Find the Value of a Line at a Specified Point or filling a table using an Equation (Substitution)

To do this, you need to be able to substitute values from a table into an equation to find the corresponding value for the second variable.

For example, you are asked to fill in a table from 0 to 5 for the equation y=3x+8, so you substitute in the values from 0 to 5 into the equation for x, as shown here:

y=(3*0)+8=8

y=(3*1)+8=11

y=(3*2)+8=14

y=(3*3)+8=17

y=(3*4)+8=20

y=(3*5)+8=23

You then put the values in a table as follows:

x y

0 8

1 11

2 14

3 17

4 20

5 23

To do this, you need to be able to substitute numbers into an equation.

For example, if you have the equation y=-2x+14 and have to find the x value where y=3, you would substitute 3 in for y, to find x, as follows:

3=-2x+14

-11=-2x

11=2x

11/2=x

5.5=x

Therefore x is 5.5 where y is 3.

Substitution into equations or formulae (using BEDMAS) - function machine NOTES

Solving linear equations - working backwards

Solving equations using a model (math playground)
Solving equations Online notes (see below for videos):
- Bitesize tutorials, Purple Math,
- More examples and practice Level 1, Level 2, Level 3
Solving equations Puzzles to print off: Puzzle1, 2 step, x on both sides, substitution
Rearranging formulae
Solving inequations (extension)

Demonstrates, step-by-step and with illustrations, how to solve one-step linear equations using addition and subtraction.www.purplemath.com/modules/solvelin.htm

Interactive solving using a model

#6 Find an Intersection Point Between Two sloping Lines to Answer a Problem

To do this, you must make two different equations equal to each other.

For example, finding the intersection between y=9x-17 and y=3x+11, you would do the following:

y=y

Therefore

9x-17=3x+11

Solve for x by getting like terms on the same sides

9x-3x=11+17

6x=28

x=28/6

x=4.67

The solve with one equation for the value of y, as follows:

y=(3*4.67)+11

y=14+11

y=25

Therefore, the point of intersection between these two equations is (4.67,25)

Finding points of intersection - simultaneous equations

Solving simultaneous equations using algebra (video), notes online, bitesize notes
- using both algebra and graphs - slides with VIDEO explanation
<extn: solving simultaneous equations by elimination>
graph match up activity

Solving Simultaneous Linear Equations .webm

Practice Tasks: Using Tables, Equations and Graphs to solve Linear Algebra Problems in CONTEXT

A possible method when approaching these types of extended linear algebra problems is listed below.

Determine what you want to know (the x and y variables, which could be things such as time and money, height and width, age and number of credits, etc)
Determine what you already know (read the situation and identify any relevant information)
Find the gradient and y-intercept for any relationships present in the problem
Form the equations for the relationships
Graph the relationships together
Compare the models to each other
Make recommendations for which model is best in different situations
Make recommendations as to how the models can be changed to be better for different situations

Pizza Problems Task, Solutions on DESMOS, video tutorial on how to make a graph - Pizza problem

Snowplanet Activity, possible Desmos solution

Linda's Party Task, possible answers, solutionondesmos

Laundry Services, possible solution

Extra practice tasks (Taxi Companies, Concrete, Internet) OR google doc version with answers.
2016 Practice Internal : Practical Printers Task, google doc version, possible Desmos solution, mark schedule

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