Algebra 1 Discovering expressions, equations and functions Overview Expressions and variables Operations in the right order Composing expressions Composing equations and inequalities Representing functions as rules and graphs. Algebra 1 Exploring real numbers Overview Integers and rational numbers Calculating with real numbers The Distributive property Square roots. Algebra 1 How to solve linear equations Overview Properties of equalities Fundamentals in solving equations in one or more steps Ratios and proportions and how to solve them Similar figures Calculating with percents.
Algebra 1 Visualizing linear functions Overview The coordinate plane Linear equations in the coordinate plane The slope of a linear function The slope-intercept form of a linear equation.
Draw a straight line through those points that represent the graph of this equation. A graph is a pictorial representation of numbered facts. There are many types of graphs, such as bar graphs, circular graphs, line graphs, and so on. You can usually find examples of these graphs in the financial section of a newspaper. Graphs are used because a picture usually makes the number facts more easily understood.
In this section we will discuss the method of graphing an equation in two variables. In other words, we will sketch a picture of an equation in two variables. All possible answers to this equation, located as points on the plane, will give us the graph or picture of the equation. A sketch can be described as the "curve of best fit. Remember, there are infinitely many ordered pairs that would satisfy the equation.
Solution We wish to find several pairs of numbers that will make this equation true. We will accomplish this by choosing a number for x and then finding a corresponding value for y. A table of values is used to record the data. In the top line x we will place numbers that we have chosen for x. Then in the bottom line y we will place the corresponding value of y derived from the equation. Of course, we could also start by choosing values for y and then find the corresponding values for x.
These values are arbitrary. We could choose any values at all. Notice that once we have chosen a value for x, the value for y is determined by using the equation. These values of x give integers for values of y. Thus they are good choices. Suppose we chose. We now locate the ordered pairs -3,9 , -2,7 , -1,5 , 0,3 , 1,1 , 2,-1 , 3,-3 on the coordinate plane and connect them with a line.
The line indicates that all points on the line satisfy the equation, as well as the points from the table. The arrows indicate the line continues indefinitely. The graphs of all first-degree equations in two variables will be straight lines. This fact will be used here even though it will be much later in mathematics before you can prove this statement.
Such first-degree equations are called linear equations. Equations in two unknowns that are of higher degree give graphs that are curves of different kinds. You will study these in future algebra courses. Since the graph of a first-degree equation in two variables is a straight line, it is only necessary to have two points. However, your work will be more consistently accurate if you find at least three points.
Mistakes can be located and corrected when the points found do not lie on a line. We thus refer to the third point as a "checkpoint. Don't try to shorten your work by finding only two points. You will be surprised how often you will find an error by locating all three points. Solution First make a table of values and decide on three numbers to substitute for x. We will try 0, 1,2. Again, you could also have started with arbitrary values of y.
The answer is not as easy to locate on the graph as an integer would be. Sometimes it is possible to look ahead and make better choices for x. We will readjust the table of values and use the points that gave integers. This may not always be feasible, but trying for integral values will give a more accurate sketch.
We can do this since the choices for x were arbitrary. How many ordered pairs satisfy this equation? Upon completing this section you should be able to: Associate the slope of a line with its steepness.
Write the equation of a line in slope-intercept form. Graph a straight line using its slope and y-intercept. We now wish to discuss an important concept called the slope of a line. Intuitively we can think of slope as the steepness of the line in relationship to the horizontal. Following are graphs of several lines. Study them closely and mentally answer the questions that follow. If m as the value of m increases, the steepness of the line decreases and the line rises to the left and falls to the right.
In other words, in an equation of the form y - mx, m controls the steepness of the line. In mathematics we use the word slope in referring to steepness and form the following definition:.
Solution We first make a table showing three sets of ordered pairs that satisfy the equation. Remember, we only need two points to determine the line but we use the third point as a check. Example 2 Sketch the graph and state the slope of. Why use values that are divisible by 3?
Compare the coefficients of x in these two equations. Again, compare the coefficients of x in the two equations. Observe that when two lines have the same slope, they are parallel. The slope from one point on a line to another is determined by the ratio of the change in y to the change in x. That is,. If you want to impress your friends, you can write where the Greek letter delta means "change in.
We could also say that the change in x is 4 and the change in y is - 1. This will result in the same line. The change in x is 1 and the change in y is 3. If an equation is in this form, m is the slope of the line and 0,b is the point at which the graph intercepts crosses the y-axis. The point 0,b is referred to as the y-intercept. If the equation of a straight line is in the slope-intercept form, it is possible to sketch its graph without making a table of values. Use the y-intercept and the slope to draw the graph, as shown in example 8.
First locate the point 0, This is one of the points on the line. The slope indicates that the changes in x is 4, so from the point 0,-2 we move four units in the positive direction parallel to the x-axis. Since the change in y is 3, we then move three units in the positive direction parallel to the y-axis.
The resulting point is also on the line. Since two points determine a straight line, we then draw the graph. Always start from the y-intercept. A common error that many students make is to confuse the y-intercept with the x-intercept the point where the line crosses the x-axis. To express the slope as a ratio we may write -3 as or. If we write the slope as , then from the point 0,4 we move one unit in the positive direction parallel to the x-axis and then move three units in the negative direction parallel to the y-axis.
Then we draw a line through this point and 0,4. Can we still find the slope and y-intercept? The answer to this question is yes. For example,. We know that each inequality in the set contains infinitely many ordered pair solutions defined by a region in a rectangular coordinate plane. When considering two of these inequalities together, the intersection of these sets defines the set of simultaneous ordered pair solutions.
When we graph each of the above inequalities separately, we have. When graphed on the same set of axes, the intersection can be determined. The intersection is shaded darker and the final graph of the solution set is presented as follows:. The graph suggests that 3, 2 is a solution because it is in the intersection.
To verify this, show that it solves both of the original inequalities:. Points on the solid boundary are included in the set of simultaneous solutions and points on the dashed boundary are not. Notice that this point satisfies both inequalities and thus is included in the solution set. Shade upper half of the line. Here point 0 , 0 satisfies the inequality, so shade the half that contains the point.
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