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Here are the three equations. Each represents the relationship between the number of games, , the number of rides, , and the dollar amount a student is spending on games and rides at a different amusement park.
Equation 1:
Equation 2:
Equation 3:
Your teacher will assign to you (or ask you to choose) 1–2 equations. For each assigned (or chosen) equation, answer the questions.
First assigned (or chosen) equation:
What’s the number of games the student could play if the student doesn’t get on any rides? On the coordinate plane, mark the point that represents this situation and label the point with its coordinates.
Second assigned (or chosen) equation:
Andre’s coin jar contains 85 cents. There are no quarters or pennies in the jar, so the jar has all nickels, all dimes, or some of each.
Linear equations can be written in different forms. Some forms allow us to better see the relationship between quantities or to predict the graph of the equation.
Suppose a person wishes to travel 7,000 meters a day by running and swimming. The person runs at a speed of 130 meters per minute and swims at a speed of 54 meters per minute.
Let represents the number of minutes of running and the number of minutes of swimming. To represent the combination of running and swimming that would allow the person to travel 7,000 meters, we can write:
We can reason that the more minutes the person runs, the fewer minutes the person has to swim to meet the goal. In other words, as increases, decreases. If we graph the equation, the line will slant down from left to right.
If the person only runs and doesn't swim, how many minutes would the person need to run?
Let's substitute 0 for to find :
On a graph, this combination of times is the point , which is the -intercept.
If the person only swims and doesn't run, how many minutes would the person need to swim?
Let's substitute 0 for to find :
On a graph, this combination of times is the point , which is the -intercept.
To determine how many minutes the person would need to swim after running for 15 minutes, 20 minutes, or 30 minutes, substitute each of these values for in the equation and find . Or, first solve the equation for :
Notice that , or , is written in slope-intercept form.
The first equation we wrote, , is a linear equation in standard form. In general, it is expressed as , where and are variables, and , and are numbers.
The two equations, and , are equivalent, so they have the same solutions and the same graph.