Linear Equations - Supplement

5/19/2018 Linear Equations - Supplement

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Lecture 1 Supplement

Week 2, Autumn 2008


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Linear equations

Recall the definition of a linear equation (or a function):none of the variables is raised to any other power than 1.

For example: 1/x + 2 = 0 is non-linearbecause x israised to the power of -1.

To solve the linear equation: 10x 2 = 0.Rearrange the expression move the xs to the left-handside and the constant to the right-hand side.

You get: 10x = 2, divide by 10

You get: x = 2/10 = 1/5.

Check by substitution: 10(1/5) 2 =0.


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Linear functions Example (1)

Consider the linear function: y = -4x + 8, which

expresses the relationshipbetween thevariables x and y.

Here x is the independentvariable and y is the

dependentvariable. To graph this function, you can (1) use a table of

values, (2) compute the intercepts (3) use the

slope and one of the intercepts.


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(1) A table of values for y = -4x+8

y = -4(-1)+8=12-1

y = -4(2)+8=02

y = -4(1)+8=41

y = -4(0)+8=80

y = -4x+8x


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(2) Computing the intercepts To compute the intercepts, set one of the variables

equal to zero and solve for the other: If y = 0, then 0 = -4x +8, or 4x = 8, or x = 2 (the x-

intercept); the coordinates for this point (2, 0)

If x = 0, then y = -4(0) + 8, and y = 8 (y-intercept);the coordinates for this point (0, 8)

Using (2, 0) and (0, 8) you can draw the graphthrough these points.


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(3) Slope and y-intercept

From y = -4x + 8, we learn that the slope is -4. Since

the slope is negative, the graph is downwardsloping.

Take the y-intercept (0, 8) and plot it into the graph.

Using the slope, move one step to the right(increase x by one unit) and four steps down (ydecreases by 4 units), you arrive at the point (1, 4),

etc. This way you find the points which are on thegraph of y = -4x + 8.


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x

y

y = -4x + 8

8

2


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Example (2)

Suppose you are given two coordinates, (x0, y0)=(1, 3)and (x1, y1)=(2, 4), and asked to derive a corresponding

linear equation.

Use the general form of the linear equation y = ax + b. You need to find a(the slope) and b(the y-intercept) to

be able to do this. Using the coordinates from above, write down:

y0 = ax0 + b => -3 = a(1) + b => b = -3 - a (1)

y1 = ax1 + b => 4 = a(2) + b => b = -2a + 4. (2)


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Example (2)

The equations (1) and (2) form a system of equations

Set -3 - a = -2a + 4 and solve for a => a = 7.

Substitute a = 7 back into one of the equations above toget: b = -3 - 7 = -10.

Hence, the linear function takes the form:y = 7x - 10.


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Example (2)

Alternatively, compute the slope and then use one of thecoordinates to compute the y-intercept (b).

The slope:

71

7

12

)3(4

01

01==

=

=

=

x

y

xx

yya

Plug into the general form either pair of values, say (1, -3):

-3 = 7(1)+b, and solve for b: b = -10

You can now write down the specific form of the linearfunction: y = 7x 10.


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x

y

y =7x - 10

-10

10/7


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System of equations

Three equations and three unknowns: Check the

Renshaw book section 3.17 p. 109.

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Linear Equations - Supplement