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5.05 Comparing linear relationships

Lesson

We've already learned how to identify linear equations, which are shown graphically as straight line graphs. Now we are going to learn how to compare the features of linear equations. Linear equations are often written in slope-intercept form which is handy because it helps us identify the slope and $y$y-intercepts of these lines as shown below.

  • Slope: we may be asked which of the two linear equations has a greater/ lesser slope. This may also be referred to as the steepness of the line, the rate of change or simply increasing or decreasing faster/slower. The slope refers to how much the dependent variable changes for every time the independent variable increases by one. In context, we might be asked questions like, which carpenter charges more per hour, who ran at a quicker pace or which is the better deal?
  • y-intercept: the $y$y-intercept is where the line crosses the $y$y axis. The line with the greater $y$y-intercept will be the one that crosses at a higher number on the $y$y axis. In context, we might be asked questions like, which has the higher start-up cost?
  • x-intercept: this is where the line crosses the $x$x axis. Just like the y-intercept, the line with the greater $x$x-intercept will be the one that crosses at a higher number on the $x$x axis. In context, we might be asked questions like, which company will run out first or who will get home first?
  • Coordinates: each pair of coordinates has an $x$x and a $y$y value $\left(x,y\right)$(x,y). We may be asked to substitute an $x$x or a $y$y value into a pair of linear equations to see which equation produces the greater/ smaller value for the other variable.

We may also be asked the compare two linear models which involve a scenario. 

Let's look through some examples that compare these different features of linear equations.

Examples

Question 1

In which of the following is $y$y increasing faster?

  1. $x$x $0$0 $1$1 $2$2
    $y$y $3$3 $10$10 $17$17
    A

    Loading Graph...

    B
    $x$x $0$0 $1$1 $2$2
    $y$y $3$3 $10$10 $17$17
    A

    Loading Graph...

    B

Question 2

Which of the following has the higher $y$y-intercept?

  1. The line with a slope of $4$4 that crosses the $y$y-axis at $\left(0,6\right)$(0,6).

    A

    The line given by the equation $y=x+4$y=x+4

    B

    The line with a slope of $4$4 that crosses the $y$y-axis at $\left(0,6\right)$(0,6).

    A

    The line given by the equation $y=x+4$y=x+4

    B

Question 3

Mario wants to determine which of two slow-release pain medications is more rapidly absorbed by the body.

For the liquid form, the amount of the medication in the bloodstream is presented in the graph below. The results for the capsule form are presented in the table below.

     
Loading Graph...
 
   
     
Time passed (mins) Amount in bloodstream (mgs)
$4$4 $24.6$24.6
$7$7 $42.3$42.3
$10$10 $60$60
$13$13 $77.7$77.7
   
  1. At what rate, in mg per minute, is the liquid form absorbed?

  2. At what rate, in mg per minute, is the capsule form absorbed?

  3. In which form is the medication absorbed more rapidly?

    In liquid form.

    A

    In capsule form.

    B

    In liquid form.

    A

    In capsule form.

    B

Outcomes

8.AF.7

Compare properties of two linear functions given in different forms, such as a table of values, equation, verbal description, and graph (e.G., Compare a distance-time graph to a distance-time equation to determine which of two moving objects has greater speed).

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