Question

a kindergarten teacher bought $21 worth of stickers and cardstock for his class. the stickers cost $1.50 a sheet and the cardstock cost $3.50 per pack. the equation 1.5s + 3.5c = 21 represents the relationship between sheets of stickers, s, packs of cardstock, c, and the dollar amount a kindergarten teacher spent on these supplies.
graph with points (0,6), (7,3), (14,0); axes: s (sheets of stickers) and c (packs of cardstock)
a. explain how we can tell that this graph represents the given equation.
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b. what do the vertical and horizontal intercepts, (0,6) and (14, 0), mean in this situation?
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Explanation:

Part a

Step1: Check Intercepts

To verify if the graph represents the equation \(1.5s + 3.5c=21\), we can check the intercepts. For the \(s\)-intercept, set \(c = 0\). Then \(1.5s=21\), solving for \(s\) gives \(s=\frac{21}{1.5}=14\), which matches the point \((14,0)\) on the graph. For the \(c\)-intercept, set \(s = 0\). Then \(3.5c = 21\), solving for \(c\) gives \(c=\frac{21}{3.5}=6\), which matches the point \((0,6)\) on the graph. Also, we can check the point \((7,3)\). Substitute \(s = 7\) and \(c = 3\) into the equation: \(1.5(7)+3.5(3)=10.5 + 10.5=21\), which satisfies the equation. So the points on the graph satisfy the given linear equation.

Step2: Verify Slope

The slope of the line from \((0,6)\) to \((14,0)\) is \(m=\frac{0 - 6}{14 - 0}=\frac{-6}{14}=-\frac{3}{7}\). Rewriting the equation \(1.5s+3.5c = 21\) in slope - intercept form (\(c=ms + b\)): \(3.5c=-1.5s + 21\), \(c=\frac{-1.5}{3.5}s+\frac{21}{3.5}\), \(\frac{-1.5}{3.5}=-\frac{3}{7}\) and \(\frac{21}{3.5} = 6\), so the slope and intercepts of the line from the graph match the equation \(1.5s + 3.5c=21\).

The vertical intercept \((0,6)\) (where \(s = 0\)) means that when the teacher buys \(0\) sheets of stickers (\(s = 0\)), the number of packs of cardstock (\(c\)) bought is \(6\) packs. This is because if no stickers are bought, all the \(\$21\) is spent on cardstock, and since each pack of cardstock costs \(\$3.50\), the number of packs is \(\frac{21}{3.5}=6\). The horizontal intercept \((14,0)\) (where \(c = 0\)) means that when the teacher buys \(0\) packs of cardstock (\(c = 0\)), the number of sheets of stickers (\(s\)) bought is \(14\) sheets. This is because if no cardstock is bought, all the \(\$21\) is spent on stickers, and since each sheet of stickers costs \(\$1.50\), the number of sheets is \(\frac{21}{1.5}=14\).

Answer:

We can tell the graph represents the equation by checking that the intercepts \((0,6)\) (when \(s = 0\), \(c = 6\)) and \((14,0)\) (when \(c=0\), \(s = 14\)) satisfy the equation \(1.5s+3.5c = 21\). Also, the point \((7,3)\) satisfies the equation, and the slope of the line (calculated from the intercepts) matches the slope of the line from the equation in slope - intercept form.

Part b