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Equations of Lines


Alignments to Content Standards: 8.EE.B

Task

The figure below shows two lines. One is described by the equation $4x-y=c$ and the other by equation $y=2x+d$, for some constants $c$ and $d$. They intersect at the point $(p,q)$.

Linear_functions_2ae8388aff8443f692a52ca53bbe6bf4

  1. How can you interpret $c$ and $d$ in terms of the graphs of the equations above?
  2. Imagine you place the tip of your pencil at point $(p,q)$ and trace line $l$ out to the point with $x$-coordinate $p+2$. Imagine I do the same on line $m$. How much greater would the $y$-coordinate of your ending point be than mine?

IM Commentary

This task requires students to use the fact that on the graph of the linear equation $y=ax+c$, the $y$-coordinate increases by $a$ when $x$ increases by one. Specific values for $c$ and $d$ were left out intentionally to encourage students to use the above fact as opposed to computing the point of intersection, $(p,q)$, and then computing respective function values to answer the question.

To see an annotated version of this and other Illustrative Mathematics tasks as well as other Common Core aligned resources, visit Achieve the Core.

Solution

  1. If we put the equation $4x-y=c$ in the form $y = 4x-c$, we see that the graph has slope 4. The slope of the graph of $y = 2x+d$ is 2. So the steeper line, $l$, is the one with equation $y=4x-c$, and therefore $-c$ is the $y$-coordinate of the point where $l$ intersects the $y$-axis. The other line, $m$, is the one with equation $y=2x+d$, so $d$ is the $y$-coordinate of the point where $m$ intersects the $y$-axis.
  2. The line $l$ has slope 4. So on $l$, each increase of one unit in the $x$-value produces an increase of 4 units in the $y$-value. Thus an increase of 2 units in the $x$-value produce an increase of $2 \cdot 4 = 8$ units in the $y$-value. The line $m$ has slope 2. So on $L_2$, each increase of 1 unit in the $x$-value produces an increase of 2 units in the $y$-value. Thus an increase of 2 units in the $x$-value produces an increase of $2\cdot2=4$ units in the $y$-value.

    Thus your $y$-value would be $8–4=4$ units larger than my $y$-value.

Jason says:

over 4 years

This is a nice, non-routine problem in y=mx+b-ology.

Since the task tells the student (via the word "respectively") that line \ell has equation 4x - y = c and line m has equation y = 2x + d, why in the solution is this a deduction? (Third sentence)

Kristin says:

over 4 years

I think too much information was given in the task statement. I've revised it.

erinondrusek says:

over 5 years

This is a very thought-provoking problem, thank you for providing it! I noticed in the commentary that it states "the y-coordinate increases by c when x increases by one". I think this is meant to be "the y-coordinate increases by a" to show that an increase of one for the input results in an increase of a, the slope, in the output.

Kristin says:

over 5 years

Thanks for catching that! I've fixed it now.