Being able to discuss numbers hypothetically – using a letter to represent a number – underpins much of this course.
Activity: Algebra Review Solving equations
Use this applet to find the kinds of equations that you need to practice most.
Use this one as a practice test.
It’s great to be able to solve equations for equations’ sake. However
in the course we encountered many problems that could be solved by setting up and solving an equation. Here are some examples. Each example also require that you understand some other area of knowledge – what an arithmetic sequence is; or what is meant by
; or how trigonometry works; or
understand the term ‘perimeter’, etc.
Solving Equations in Context
Printable: Solving Equations in Context Solutions: Solving Eq in Context Solution
1. This right angled triangle has one side length 9 cm, and one angle 18°.
From trigonometry, we know that
. Solve this equation to find the length of the side marked
cm.
2. This right angled triangle has one side marked 9 cm, and an angle 24°.
From trigonometry, we know that
. Solve this equation to find the length of the side marked
cm.
3. An
arithmetic sequence has general term
. The number 61 is part of this sequence. What term is the number 61? Set up an equation and solve it.
4. An arithmetic sequence has general term
. The number 31 is a term in this sequence. What term is the number 31? Set up an equation and solve it.
5. The arithmetic sequences
and
have a common term – that is, for some value of
,
. Set up an equation to find the value of
, then find the value of the term.
6. The arithmetic sequences
and
do not share a common term. How can this be proved?
7. The rectangle and the square shown below have the same perimeter. The sides of the rectangle and the side length of the square are given in terms of an unknown number
. Set up an equation to find the value of
. Then find the dimensions of each shape, and the perimeter.
8. A rectangle has perimeter 27.6 cm. The length is 5 times as long as the width. Set up an equation with this information and solve it to find the length and width of the rectangle.
9. A function is defined as
. Find
when
.
10. A function is defined as
. Find the value of
when
.
11. A straight line has equation
. Set up an equation to calculate the
intercept of the line.
12. A straight line has
equation
. Set up an equation to calculate the
intercept.
13. A straight line has equation
. Calculate the
intercept and the
intercept.
14. The straight lines
and
have a point in common – a point where the lines intersect each other. Set up an equation to find the
value of this point. Calculate the
value.
15. The straight lines
and
have a point in common. Set up an equation to find the
value of this point. Calculate the
value.
How to write an absolute value equation?
More generally, the form of the equation for an absolute value function is y=a| x−h |+k.
How to solve absolute value inequalities?
To solve an absolute value inequality involving “less than,” such as |X|≤p, replace it with the compound inequality −p≤X≤p and then solve as usual. To solve an absolute value inequality involving “greater than,” such as |X|≥p, replace it with the compound inequality X≤−p or X≥p and then solve as usual.
What makes a linear equation?
A linear equation only has one or two variables. No variable in a linear equation is raised to a power greater than 1 or used as the denominator of a fraction. When you find pairs of values that make a linear equation true and plot those pairs on a coordinate grid, all of the points lie on the same line.