Look for a pattern in the first three equations. Then, fill in the missing numbers in the rest of the series of equations. 1= 1 · 2⁄2
1 + 2 = 2 · 3⁄2
1 + 2 + 3 = 3 · 4⁄2
1 + 2 + 3 + ? = ? · ?⁄?
1 + 2 + 3 + ? + ? = ? · ?⁄?

Solve the following equations.
a. 11w = 88
b. 7u = 210
c. v ÷ 81 = 9
d. c ÷ 4 = 16
For the following problems, tell what operation(s) you need to use. Then, write and solve an equation for each problem.

Bill bought a computer and a printer for a total of $650. The computer cost $525. How much did he pay for the printer alone? (Let p stand for the cost of the printer.)
Bonnie loves to order products online. She ordered 27 products from one company, but unfortunately, some items were back-ordered, so she received only 19 items. How many items were missing from her order? (Let m stand for the number of missing items.)
Terri bought four lipsticks for a total cost of $20. Assuming each lipstick cost the same amount, how much did one lipstick cost? (Let l stand for the cost of one lipstick.)
Brian is conducting surveys for a research paper. Each survey takes him about 10 minutes to complete, so he can get 6 done per hour. At this rate, how many hours will it take Brian to complete 30 surveys? (Let h stand for the number of hours.)
Andy played golf three times during his vacation. His scores were 92, 87, and 88. What was his average golf score during his vacation? (Let A stand for the average.)
Tickets to a local basketball team’s games cost $12 each. Brendon and Ed each bought tickets to four games. How much did they spend on tickets altogether? (Let t stand for the amount spent on tickets.)

answer: learning objective(s)

·         describe the creation and use of systems of equations.

·         graph a system of linear equations on the coordinate plane and identify its solution.

introduction

sometimes graphing a single linear equation is all it takes to solve a mathematical problem. other times, one line just doesn’t do it, and a second equation is needed to model the situation. this is often the case when a problem involves two variables. solving these kinds of problems requires working with a system of equations, which is a set of two or more equations containing the same unknowns.

let’s take a look at systems of equations, and see what the graphs of individual equations within a system reveal about the mathematical relationship of the variables.

systems of equations

a system of equations contains two or more linear equations that share two or more unknowns. to find a solution for a system of equations, we must find a value (or range of values) that is true for all equations in the system.

the graphs of equations within a system can tell us how many solutions exist for that system. look at the images below. each show two lines that make up a system of equations (in the graph on the right the two lines are superimposed and look like a single line). how many points in common does each system of lines reveal?

one solution

no solutions

infinite solutions

if the graphs of the equations intersect, then there is one solution that is true for both equations.

if the graphs of the equations do not intersect (for example, if they are parallel), then there are no solutions that are true for both equations.

if the graphs of the equations are the same, then there are an infinite number of solutions that are true for both equations.

remember, the graph of a line represents every point that is a possible solution for the equation of that line. so when the graphs of two equations cross, the point of intersection lies on both lines, meaning that it is a possible solution for both equations. when the graphs of two equations never touch, there are no shared points and there are no possible solutions for the system. when the graphs of two equations lie on top of one another, they share all their points and every one is a possible solution.

graphing as a solution method

graphing equations in order to identify a specific point of intersection is usually not a precise way to solve systems because it is often difficult to see exactly where two lines intersect (unless you are using a computer-based graphing program that allows you to zoom in on a point). however, the graph of a system of equations can still give a good idea of what type of solution, if any, exists. let’s graph a system, and see how it works.

graph the system y = 3x and x + 2y = 4. how many solutions does this system have?

a graph of the two lines y = 3x and x + 2y = 4 shows us that the lines intersect, meaning that there is a single (x, y) value that satisfies both equations. looking at the graph does not tell us exactly where that point is, but we don’t need to know that information here, because the question only asks for the number of solutions.

so a system made of two intersecting lines has one solution. now let’s look at a different situation:

how many solutions exist for the system y -0.5x = 7 and 2y = x − 3?

plotting both equations, it looks like there is no solution—the lines are parallel. to check this finding, we can compare the slopes of the equations. to make that easy, we’ll rearrange the equations into slope-intercept form, or y = mx + b.   that gives us the equations y = 0.5x + 7 and y = 0.5x − 1.5.   yes, the slope of both lines is 0.5, which means the lines really are parallel. they never intersect, so there is no point that lies on both lines, and no solution to the system.

step-by-step explanation:

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