5. An open equation is an equation with one or more variables that may be true or false depending on the values of the variables. An example of and open equation is [tex] 4x+3=7 [/tex].
- Notice that if [tex] x=1 [/tex], then
[tex] 4x+3=7 [/tex]
[tex] 4(1)+3=7 [/tex]
[tex] 7=7 [/tex]
As you can see, when [tex] x=1 [/tex], the open equation [tex] 4x+3=7 [/tex] is true.
- Let's check what happen if [tex] x=3 [/tex]:
[tex] 4x+3=7 [/tex]
[tex] 4(3)+3=7 [/tex]
[tex] 12+3=7 [/tex]
[tex] 15\neq 7 [/tex]
Since 15 is not equal to seven, when [tex] x=3 [/tex], the open equation [tex] 4x+3=7 [/tex] is false.
6. Equations can be used as models to represent how a particular part of the real world works. Since model equations use variables to relate real world quantities, you can use equations to make predictions. For instance, suppose you have the equation [tex] y=0.50x [/tex]
where [tex] y [/tex] is the total cost (in dollars) for parking for [tex] x [/tex] hours.
You can predict, using the equation, how much we are going to pay for 2 hours of parking.
[tex] y=0.50x [/tex]
[tex] y=0.50(2) [/tex]
[tex] y=1.00 [/tex]
Using an equation, we predicted that parking 2 hours will cost $1.
7. Let [tex] y [/tex] be the total cost of the pizza, and let [tex] x [/tex] be the number of slices. We know from our problem that one slice costs 1.50, so x slices will cost [tex] 1.50x [/tex]. Now the only thing left is relate the total cost [tex] y [/tex] and the cost of x slices [tex] 1.50x [/tex] using an equal sign to create and equation to model the situation:
[tex] y=1.50x [/tex]
Notice that we can predict how much is going to cost a pizza of 8 slices just by replacing [tex] x [/tex] with 8 in our pizza equation
[tex] y=1.50x [/tex]
[tex] y=1.50(8) [/tex]
[tex] y=12 [/tex]
