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, if we have $x$ pounds of flour that costs \$0.80 per pound and $y$ pounds of sugar that costs \$0.50 per pound, and the total cost is \$9.00, then we can write an equation like this to represent the relationship between $x$ and $y:$
$$0.8x + 0.5y = 9$$
Since 5 pounds of flour costs \$4.00 and 10 pounds of sugar costs \$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 two points shown that are not on the line. What do they mean in the context? The point $(1,14)$ means that there is 1 pound of flour and 14 pounds of sugar. The total cost for this is $0.8 \boldcdot 1 + 0.5 \boldcdot 14$ or \$7.80. Since the cost is not \$9.00, this point is not on the graph. Likewise, 9 pounds of flour and 16 pounds of sugar costs $0.8 \boldcdot 9 + 0.5 \boldcdot 16$ or \$15.20, so the other point is not on the graph either.
Suppose we also know that the flour and sugar 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 two of the three labeled points:
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 two lines in the coordinate plane,
- The coordinates of a point that is on both lines makes both equations true.
- The coordinates of a point on only one line makes only one equation true.
- The coordinates of a point on neither line make both equations false.