We studied linear relationships in an earlier unit. We learned that values of $x$ and $y$ that make an equation true correspond to points $(x,y)$ on the graph.

For example, let’s plan the base rocks for a terrarium. We have $x$ pounds of river rocks that cost $0.80 per pound and $y$ pounds of unpolished rocks that cost $0.50 per pound, and the total cost is $9.00, so we can write an equation like this to represent the relationship between $x$ and $y:$ $0.8x + 0.5y = 9$

Because 5 pounds of river rocks cost $4.00 and 10 pounds of unpolished rocks cost $5.00, we know that $x=5$, $y=10$ is a solution to the equation, and the point $(5,10)$ is a point on the graph.

The line shown is the graph of the equation. Notice that there are 2 points shown that are not on the line. What do they mean in the context?

<p>The graph of a line in the x y plane. The line slants downward and right, crosses the y axis at 18, and passes through the point 5 comma 10. Two additional points, 9 comma 16 and 1 comma 14, are labeled on the graph.</p>  

The point $(1,14)$ means that there is 1 pound of river rock and 14 pounds of unpolished rocks. The total cost for this is $0.8 \boldcdot 1 + 0.5 \boldcdot 14$ or $7.80. Because the cost is not $9.00, this point is not on the line. Likewise, 9 pounds of river rocks and 16 pounds of unpolished rocks cost $0.8 \boldcdot 9 + 0.5 \boldcdot 16$ or $15.20, so the other point is not on the line either.

Suppose we also know that the river rocks and unpolished rocks together weigh 15 pounds. That means that $x+y=15$.

If we draw the graph of this equation on the same coordinate plane, we see it passes through 2 of the 3 labeled points:

<p>The graph of two intersecting lines in the x y plane. The first line slants downward and right, crosses the y axis at 18, and passes through the point 5 comma 10. The second line slants downward and to the right and passes through the points 1 comma 14 and 5 comma 10. An additional point, 9 comma 16, is labeled on the graph.</p>  

The point $(1,14)$ is on the graph of $x+y=15$ because $1 + 14 = 15$. Similarly, $5 + 10 = 15$. But $9 + 16 \neq 15$, so $(9, 16)$ is not on the graph of $x+y = 15$.

In general, if we have 2 lines in the coordinate plane and we have their corresponding equations,

• The coordinates of a point on a line make that equation true.
• The coordinates of a point off of a line make that equation false.
• The coordinates of a point that is the intersection of the 2 lines make both equations true.