Solve Systems of Equations by Graphing Quiz

PDF page 586: Solutions of a system of equations are the values of the variables that make all the equations true. A solution of a system of two linear equations is represented by an ordered pair (x, y). To determine if an ordered pair is a solution to a system of two equations, we substitute the values of the variables into each equation. If the ordered pair makes both equations true, it is a solution to the system. Let’s consider the system below: ⎧3x − y = 7

PDF page 586: Solutions of a system of equations are the values of the variables that make all the equations true. A solution of a system of two linear equations is represented by an ordered pair (x, y). To determine if an ordered pair is a solution to a system of two equations, we substitute the values of the variables into each equation. If the ordered pair makes both equations true, it is a solution to the system. Let’s consider the system below: ⎧3x − y = 7

PDF page 586: ⎩x − 2y = 6 A linear equation in two variables, like 2x + y = 7, has an infinite number of solutions. Its graph is a line. Remember, every point on the line is a solution to the equation and every solution to the equation is a point on the line. To solve a system of two linear equations, we want to find the values of the variables that are solutions to both equations. In other words, we are looking for the ordered pairs (x, y) that make both equations true. These are called the solutions to a system of equations.

PDF page 586: The ordered pair (2, −1) made both equations true. Therefore (2, −1) is a solution to this system. Let’s try another ordered pair. Is the ordered pair (3, 2) a solution?

PDF page 587: The ordered pair (3, 2) made one equation true, but it made the other equation false. Since it is not a solution to both equations, it is not a solution to this system.

PDF page 587: In this chapter we will use three methods to solve a system of linear equations. The first method we’ll use is graphing. The graph of a linear equation is a line. Each point on the line is a solution to the equation. For a system of two equations, we will graph two lines. Then we can see all the points that are solutions to each equation. And, by finding what the lines have in common, we’ll find the solution to the system. Most linear equations in one variable have one solution, but we saw that some equations, called contradictions, have no solutions and for other equations, called identities, all numbers are solutions. Similarly, when we solve a system of two linear equations represented by a graph of two lines in the same plane, there are three possible cases, as shown in Figure 5.2:

PDF page 590: Step 1. Graph the first equation. Step 2. Graph the second equation on the same rectangular coordinate system. Step 3. Determine whether the lines intersect, are parallel, or are the same line. Step 4. Identify the solution to the system.

PDF page 590: ◦ If the lines are parallel, the system has no solution.

PDF page 590: Graph the two lines. Determine the point of intersection. The lines intersect at (1, 3).

PDF page 590: The solution is (1, 3).

PDF page 591: Determine the point of intersection. The lines intersect at (−1, 2). 3x + y = −1 Check the solution in both equations. 2x + y = 0

PDF page 591: The solution is (−1, 2).

PDF page 596: If you write the second equation in Example 5.8 in slope-intercept form, you may recognize that the equations have the same slope and same y-intercept. When we graphed the second line in the last example, we drew it right over the first line. We say the two lines are coincident. Coincident lines have the same slope and same y-intercept.

PDF page 596: Coincident lines have the same slope and same y-intercept.

PDF page 597: A consistent system of equations is a system of equations with at least one solution. An inconsistent system of equations is a system of equations with no solution. We also categorize the equations in a system of equations by calling the equations independent or dependent. If two equations are independent equations, they each have their own set of solutions. Intersecting lines and parallel lines are independent. If two equations are dependent, all the solutions of one equation are also solutions of the other equation. When we graph two dependent equations, we get coincident lines.