Straight Line Equations and Inequalities

Topics: Linear equation, Line Pages: 6 (1871 words) Published: July 22, 2013
Straight Line Equations and Inequalities
A: Linear Equations - Straight lines
Please remember that when you are drawing graphs you should always label your axes and that y is always shown on the vertical axis. A linear equation between two variables x and y can be represented by y = a + bx where “a” and “b” are any two constants. For example, suppose we wish to plot the straight line If x = -2, say, then y = 3 + 2(-2) = 3 - 4 = -1 If x= -2 -1 -1 1 0 3 1 5 2 7 As you can see, we have plotted the five points on the graph. They do indeed all lie on a straight line and we have joined them together to show the line. Of course, you could draw the line by just plotting any two points on it and then joining and extending those two points. y = 3 + 2x ..... and so on (see table below)

Then y =

y

x

The equation simply represents the relationship between two variables x and y. For example: suppose our basic salary is £4000 and we add commission to that at the rate of 5% of our total sales. Call y our total salary and call x our sales (both in £) then we could represent this relationship as y = 4000 + 0.05x (5% is five hundredths i.e. 0.05) Then, if we knew that total sales were 6000, we could work out total salary: y = 4000+0.05(6000) or £4300

For our next example, we will draw the equation y = 6 - x on a graph (using just two points): For y = 6 - x Put x = 1 then y = 6 - 1 = 5 Put x = 4 then y = 6 - 4 = 2

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y

Y=6 - X

Plot the two points x=1, y=5 x=4, y=2 on the graph ... and then join and extend. We have now drawn the line. See the exercise below. x

EXERCISE 1: Draw the following on the same graph: y = 1 + 4x and y = 6 - 2x

B: Simultaneous Equations
When we have two linear equations in two variables, say x and y, then we can often find a “unique solution” for these equations. That is, we can often find a value for x and a value for y which “satisfies” both equations simultaneously. On a graph, it is the point where the two lines “cross” each other. For example, in economics when we have a demand curve and a supply curve then market equilibrium would be where supply equals demand (where the two lines cross). Example: Take the two equations 2x + 3y = 12 and x - y = 1 We will “solve” these two equations using three alternative methods: (a) simultaneous equations, (b) substitution and (c) graphically. By the way:- when we use a bracketed number, such as(ii), we mean equation two. When we write (ii)*3 we mean that you should multiply every term in equation two by 3. Method (a) - Simultaneous equations: 2x + 3y = 12 x -y = 1 2x - 2y = 2 5y = 10 (i) (ii) (iii) = (ii)*2 (iv) = (i) minus (iii)

hence, y = 2

Now, put y = 2 in to equation (ii), say: x - 2 = 1 or x = 3 Solution: y = 2 and x = 3

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Method (b) - Substitution: You may prefer this method: 2x + 3y = 12 x - y= 1 (i) (ii)

By re-arranging equation (ii) we can see that x = 1 + y Now use this expression to replace x in equation (i) which becomes: 2(1+y) + 3y = 12 or 2 + 2y + 3y = 12 or 2 + 5y = 12 Hence, 5y = 12 - 2 and since x = 1 + y or 5y = 10 then x = 3 so that y = 2 ...... and we have a solution: x = 3, y = 2.

Method (c) - The Graphical solution: y As an exercise you should verify that the graph is drawn correctly. (For equation (i), put x = 0 then y = 4 and put x = 6 then y = 0 etc.) x- y = 1 Visually, you can see that the two lines “intersect” at x = 3 and y = 2. x

2x + 3y = 12

By the way, normally we would not recommend the graphical solution ...you will know why when you see some of your fellow students (and my) graphs! See you (and your graphs) in class. Also – it takes far too long. As a final point in this section: what would happen if your two lines were parallel? Where identical? Think of a graph in both cases (and see the exercise below). EXERCISE 2 - Solve the following equations (answers at rear): 1. y = 5 + 2x and y = 7x 2. x + y = 5 and 2x - y = 7 3. 2x + 3y = 13 and 4x + y = 11 4. x +y = 0 and 4x +...
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