Download Linear Equations

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Helpsheet
Giblin Eunson Library
LINEAR EQUATIONS
Use this sheet to help you:
• Solve linear equations containing one unknown
• Recognize a linear function, and identify its slope and intercept parameters
• Recognize and solve linear simultaneous equations with two unknowns by
both algebraic and graphical methods
Author: Carter, D.,
Design and Layout: Pesina, J.
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LINEAR EQUATIONS
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Operations on an equation
Any operation can be performed on an equation, as long as the operation is performed
on the whole of both sides of the equation.
e.g. 1. x + 7 = 12
subtract 7 from both sides, which gives
→ x + 7 – 7 = 12 – 7, which simplifies to
→ x=5
2.
multiply both sides by 5, which gives
→
which simplifies to
→ x = 30
3.
subtract 7 from both sides, which gives
→
→
,multiply both sides by x
→ x + 5 = 3x, subtract x from both sides
→ x – x +5= 3x – x
→ 5 = 2x, divide both sides by 2
→
→
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LINEAR EQUATIONS
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Variables and Parameters
The known values in an equation are called constants, coefficients or parameters.
Sometimes when discussing an equation, we may want to look at it in a general
way, without specifying values for the parameters, so we assign letters rather than
numbers. By convention we usually assign letters from the beginning of the alphabet
to distinguish them from the variables (unknowns), which are usually assigned letters
from the end of the alphabet.
Sometimes subscripts or superscripts are used as part of the labelling of variables
or parameters For example qD and qS, where the superscripts D and S distinguish
quantity demanded and quantity supplied. Similarly we might write t0 and t1 where the
subscripts 0 and 1 distinguish an initial tax rate from the new tax rate.
The linear equation and its solution
Any linear equation with one unknown has the general form
ax + b = c
where x is the variable and a, b and c are the unspecified parameters (with a ≠ 0). Its
solution is
Linear functions
y = ax + c is an example of a linear equation with two variables, x and y, and two
parameters, a and c. A linear equation which contains two variables is called a linear
function. A way of denoting a function is y = f(x) or y = g(x) or y = φ(x).
The variable which appears on the right hand side of the equation (x) is called the
independent variable and the variable on its own on the left hand side of the equation
(y) is called the dependent variable.
Graphs of linear functions
The independent variable, x, is measured along the horizontal axis, and the dependent
variable, y, is measured along the vertical axis. The point where the two axes meet is
called the origin.
Every point on the x-y plane corresponds to a unique pair of values of x and y. These
values are known as the coordinates of the point.
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LINEAR EQUATIONS
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The slope and y-intercept of a linear function
The slope or gradient of a graph can be measured looking at the increase in y that
results when x increases by one unit. When given a linear function in the form y = ax
+ c, the slope or gradient is a. If a is positive the graph slopes upwards from left to right
and if a is negative, the graph slopes downward from left to right.
If a = 0, then ax = 0, which leaves y = b. The resulting graph is a horizontal line, with an
intercept of b on the y-axis.
The equation x = a, results in a vertical line graph, with an x-intercept of a on the x axis.
The y-intercept (the point where the line cuts the y axis) is found at the point where x =
0, therefore when given a linear function in the form y = ax + c, the y-intercept is c.
(To find the x-intercept, y = 0).
A linear function may also be given in the form ax + by + c = 0.
Simultaneous linear equations
If two equations, such as y = 4x and 2x + y = 6, are both true at the same time, they are
called simultaneous equations. What is needed here is to find the pair of values (x, y)
that satisfies both of these equations. This is what is meant by ‘solving simultaneous
equations’. In the example given we can substitute the value for y (4x) from the first
equation into the second equation to get 2x + 4x = 6 → 6x = 6 → x = 1. Once you have
found a value for x, you can substitute that value into one of the equations to find the
value for y. Substituting into the first equation gives us y = 4 x 1 → y = 4. So the pair of
values is (1, 4).
There are other techniques for solving simultaneous equations, which you may have
learned previously. You may prefer your method to the one shown.
Graphical solution of simultaneous equations
The solution to a pair of simultaneous linear equations is given by the coordinates of
the point of intersection of their graphs.
If the graphs are parallel i.e. they have the same gradient (slope), they will never
intersect and therefore will not have a solution.
If you are given three or more equations with two unknowns, unless there is unique
point of intersection for the equations there will be no pair of values that satisfy the
equations simultaneously.
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LINEAR EQUATIONS
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Three linear equations with three unknowns
Solving three linear equations with three unknowns is an extension of the method
used to solve two simultaneous equations.
e.g.
z = x + y + 4
(1)
z = 2x – y
(2)
z = 3x – 4y
(3)
Step 1: let (1) = (2)
x + y + 4 = 2x – y
→ x = 2y + 4
(4)
Step 2: let (2) = (3)
2x – y = 3x – 4y
→ x = 3y
(5)
Step 3: let (4) = (5)
2y + 4 = 3y
→ y=4
Step 4: Substitute y = 4 into (5)
x = 3(4)
→ x = 12
Step 5: Substitute y = 4 and x = 12 into (1) (* could substitute into (1), (2) or (3))
z = 12 + 4 + 4
→ z = 20
So our solution is x = 12, y = 4 and z = 20.
You can check in the other two equations to see that this is the correct solution.
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