Linear Algebra
Objectives
NA5-7 Form and solve linear equations. NA6-5 Form and solve linear equations and inequations and simultaneous equations with two unknowns. NA6-7 Relate graphs, tables, and equations to linear, relationships found in number and spatial patterns. NA6-8 Relate rate of change to the gradient of a graph |
In order to be successful, students need to be solving problems with methods related to:
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Achievement
Apply linear algebra in solving problems.
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Achievement with Merit
Apply linear algebra, using relational thinking, in solving problems.
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Achievement with Excellence
Apply linear algebra, using extended abstract thinking, in solving problems.
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The Linear Algebra Assessment is Internal Achievement Standard 91029 - 3 credits
Solving Linear Equations
A linear equation is an equation that has the highest power being 1.
When solving a linear equation, the variable must be isolated. We are finished when the variable equals a number.
You can check your answer by substituting back into the original equation.
Look at the steps and examples below and click on the links and videos for more practice.
When solving a linear equation, the variable must be isolated. We are finished when the variable equals a number.
You can check your answer by substituting back into the original equation.
Look at the steps and examples below and click on the links and videos for more practice.
Forming Equations
Part of this unit is being able to read a word problem, write an equation to represent the problem and solve it.
The following chart shows some words that may be used to represent the different mathematical operations.
Click on this link for practice : http://www.mathgoodies.com/lessons/vol7/equations.html
The following chart shows some words that may be used to represent the different mathematical operations.
Click on this link for practice : http://www.mathgoodies.com/lessons/vol7/equations.html
Linear Inequations
An inequation is solved using the same steps as when solving an equation, with one exception:
If a step involves multiplication or division by a negative number, the sign changes direction (< changes to > and > changes to <)
If a step involves multiplication or division by a negative number, the sign changes direction (< changes to > and > changes to <)
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Substituting Into a Formulae
In this section, you will practice substituting a number(s) into a formula for a variable(s). The expression will then need to be simplified or the equation will then need to be solved.
Check out these websites for more examples and some practice
https://www.mathsisfun.com/algebra/substitution.html http://www.cimt.plymouth.ac.uk/projects/mepres/book8/bk8i12/bk8_12i1.htm |
Graphing Lines
A) Graphing by Hand:
We can graph the line by hand by creating a table of values..
To create a table of values, choose at least 3 values for x and substitute them into the equation to get the three corresponding y - values.
We can graph the line by hand by creating a table of values..
To create a table of values, choose at least 3 values for x and substitute them into the equation to get the three corresponding y - values.
Use the "gradient y-intercept form": y = mx + c, where m is the gradient and c is the y - intercept.
Start by plotting the y - intercept (c) and count using the gradient (rise / run). Plot another point and join the two points.
Start by plotting the y - intercept (c) and count using the gradient (rise / run). Plot another point and join the two points.
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B) Using a graphics calculator:
1) Go the the graphs menu
2) Enter the equation into the calculator. It must be in the form y = mx + c
3) Go back to the menu and choose the table menu
4) Plot the points off the table.
1) Go the the graphs menu
2) Enter the equation into the calculator. It must be in the form y = mx + c
3) Go back to the menu and choose the table menu
4) Plot the points off the table.
Solving Simultaneous Equations
Solving simultaneous equations means solving two equations for two unknowns. This can be done either using algebra or a graphics calculator.
A) Using Algebra: Substitution Method or Elimination Method
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Check out the websites for some more help on solving simultaneous equations:
http://revisionmaths.com/gcse-maths-revision/algebra/simultaneous-equations
http://www.themathpage.com/alg/simultaneous-equations.htm
http://revisionmaths.com/gcse-maths-revision/algebra/simultaneous-equations
http://www.themathpage.com/alg/simultaneous-equations.htm
Graphing Linear Inequalities
Start by graphing the inequality as you would an equation. See the section above on graphing lines if you need to review this.
If the inequality contains an equals sign (eg. greater than or equal to), then the line is solid line.
If the inequality contains just a less than or greater than sign, then the line should be dotted.
Once the line is graphed, look at the inequality and shade the correct side. If y is greater than expression, shade above. If y is less than the expression than shade below.
If the inequality contains an equals sign (eg. greater than or equal to), then the line is solid line.
If the inequality contains just a less than or greater than sign, then the line should be dotted.
Once the line is graphed, look at the inequality and shade the correct side. If y is greater than expression, shade above. If y is less than the expression than shade below.
Check out these links for more practice.
http://www.purplemath.com/modules/ineqgrph.htm
http://www.mathsisfun.com/algebra/graphing-linear-inequalities.html
http://www.purplemath.com/modules/ineqgrph.htm
http://www.mathsisfun.com/algebra/graphing-linear-inequalities.html