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Linear Equations
A linear equation looks like any other equation. It is made up of two expressions set equal to each other. A linear equation is special because:
A linear equation in two variables describes a relationship in which the value of one of the variables depends on the value of the other variable. In a linear equation in x and y, x is called x is the independent variable and y depends on it. We call y the dependent variable. If the variables have other names, yet do have a dependent relationship, the independent variable is plotted along the horizontal axis. Most linear equations are functions (that is, for every value of x, there is only one corresponding value of y). When you assign a value to the independent variable, x, you can compute the value of the dependent variable, y. You can then plot the points named by each (x,y) pair on a coordinate grid. The real importance of emphasizing graphing linear equations with your students, is that they should already know that any two points determine a line, so finding many pairs of values that satisfy a linear equation is easy: Find two pairs of values and draw a line through the points they describe. All other points on the line will provide values for x and y that satisfy the equation.
Describing Linear Relationships
Examples of Linear Relationships
 timeIn this equation, for any given steady rate, the relationship between distance and time will be linear. However, distance is usually expressed as a positive number, so most graphs of this relationship will only show points in the first quadrant. Notice that the direction of the line in the graph below is from bottom left to top right. Lines that tend in this direction have positive slope. A positive slope indicates that the values on both axes are increasing from left to right.
timeIn this equation, since you won't ever have a negative amount of water in the bucket, the graph will also show points only in the first quadrant. Notice that the direction of the line in this graph is top left to bottom right. Lines that tend in this direction have negative slope. A negative slope indicates that the values on the y axis are decreasing as the values on the x axis are increasing.
Again in this graph, we are relating values that only make sense if they are positive, so we show points only in the first quadrant. In this case, since no polygon has fewer than 3 sides or angles, and since the number of sides or angles of a polygon must be a whole number, we show the graph starting at (3,3) and indicate with a dashed line that points between those plotted are not relevant to the problem.
Since it's perfectly reasonable to have both positive and negative temperatures, we plot the points on this graph on the full coordinate grid. |
Slope
Two-step Linear Equations with Rational Numbers. |
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Slope
The slope of a line tells two things: how steep the line is with respect to the y axis and whether the line slopes up or down when you look at it from left to right. When you're plotting data, slope tells you the rate at which the dependent variable is changing with respect to the change in the independent variable. This gives you a valuable clue about how to find slope: Pick any two points on the line. To find how fast y is changing, subtract the y value of the second point from the y value of the first point (y2 – y1). To find how fast x is changing, subtract the x value of the second point from the x value of the first point (x2 – x1). To find the rate at which y is changing with respect to the change in x, write your results as a ratio: (y2 – y1)/(x2 – x1).
 2 – 4)/(1 – 2) = 6/3. or 2. It does not matter which point you
designate as point 1, just as long as you use the same point
as the first point when calculating change in y and change in
x. If we designate Point B as the first
point and Point A as the second point, the value of
the slope is the same: (4 – 2)/(2 – 1) = 6/3, or 2. It is also the same value
you will get if you choose any other pair of points on the line
to compute slope.
The equation of a line can be written in a form that gives
away the slope and allows you to draw the line without any
computation. If students are comfortable with solving a
simple two-step linear equation, they can write linear
equations in slope-intercept form. The slope-intercept form
of a linear equation is y = mx +
b. In the equation, x and y are the
variables. The numbers m and
b give the slope of the line (m) and the
value of y when x is 0 (b). The value of
y when x is 0 is called the y-intercept
because (0,y) is the point at which the line crosses the
y axis. You can draw the line for an equation in this
form by plotting (0,b), then using m to find
another point. For example, if m is 1/2, count
 2 change in y,
there is a 1 change in x.
Now look at b in the equation: 3 should be the y value where x
= 0 and it is.
When a line slopes up from left to right, it has a positive slope. This means that a positive change in y is associated with a positive change in x. The steeper the slope, the greater the rate of change in y in relation to the change in x. When you are dealing with data points plotted on a coordinate plane, a positive slope indicates a positive correlation and the steeper the slope, the stronger the positive correlation. Consider gas mileage. If you drive a big, heavy, old car, you get poor gas mileage. The rate of change in miles traveled is low in relation to the change in gas consumed, so the value m is a low number and the slope of the line is fairly gradual. If you drive a light, efficient car, you get better gas mileage. The rate of change in the number of miles you travel is higher in relation to the change in gas consumed, so the value of m is a greater number and the line is steeper. Both rates are positive, because you still travel a positive number of miles for every gallon of gas you consume. |
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When a line slopes down from left to right, it has a negative slope. This means that a negative change in y is associated with a positive change in x. When you are dealing with data points plotted on a coordinate plane, a negative slope indicates a negative correlation and the steeper the slope, the stronger the negative correlation. Consider working in your vegetable garden. If you have a flat of 18 pepper plants and you can plant 1 pepper plant per minute, the rate at which the flat empties out is fairly high, so the absolute value of m is a greater number and the line is steeper. If you can only plant 1 pepper plant every 2 minutes, you still empty out the flat, but the rate at which you do so is lower, the absolute value of m is low, and the line is not as steep.
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When there is no change in y as x changes, the graph of the line is horizontal. A horizontal line has a slope of zero.
When there is no change in x as y changes, the graph of the line is vertical. You could not compute the slope of this line, because you would need to divide by 0. These lines have undefined slope.
Lines with the same slope are either the same line, or parallel lines.
Solving Two-Step Linear Equations with Rational Numbers When a linear equation has two variables, as it usually does, it has an infinite number of solutions. Each solution is a pair of numbers (x,y) that make the equation true. Solving a linear equation usually means finding the value of y for a given value of x.
If the equation is already in the form Two-step equations involve finding values for expressions that have more than one term. The terms in an expression are separated by addition or subtraction symbols. 2x is an expression with one term. 2x + 6 has two terms. To find a value for y given a value for x, substitute the value for x into the expression and compute. First, find the value of the term that contains x, then find the value of the entire expression.
Consider the equation
When a linear equation is not in slope-intercept form
If a = b, then a + c = b + c Consider 2x + y – 6 = 0. This equation is not in slope-intercept form. There are two ways to put it in slope-intercept form.
The two equations,
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1.
6 – 2x = y