Graphing Systems of Linear Inequalities Quiz

PDF page 586: When two or more linear equations are grouped together, they form a system of linear equations. We will focus our work here on systems of two linear equations in two unknowns. Later, you may solve larger systems of equations. An example of a system of two linear equations is shown below. We use a brace to show the two equations are grouped together to form a system of equations. ⎧2x + 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: 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 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 590: ◦ If the lines intersect, identify the point of intersection. Check to make sure it is a

PDF page 590: solution to both equations. This is 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: Coincident lines have the same slope and same y-intercept.

PDF page 596: Parallel lines have the same slope but different y-intercepts. So, if we write both equations in a system of linear equations in slope–intercept form, we can see how many solutions there will be without graphing! Look at the system we solved in Example 5.7.

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.

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.