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 that the kindergarten teacher spent on these supplies. a. explain how we can tell that this graph represents the given equation. b. what do the vertical and horizontal intercepts, (0,6) and (14,0), mean in this situation?

Explanation:

Step1: Understand the graph and equation relationship

The equation \(1.5s + 3.5c=21\) represents a linear relationship. A graph of a linear equation is a straight - line. The given graph has points that satisfy the equation \(1.5s + 3.5c = 21\). For example, if we substitute \(s = 14\) and \(c = 0\) into the equation: \(1.5\times14+3.5\times0=21 + 0=21\). If we substitute \(s = 7\) and \(c = 3\) into the equation: \(1.5\times7+3.5\times3=10.5 + 10.5=21\). If we substitute \(s = 0\) and \(c = 6\) into the equation: \(1.5\times0+3.5\times6=0 + 21=21\).

Step2: Interpret the vertical and horizontal intercepts

• Vertical intercept \((0,6)\):

When \(s = 0\) (no sheets of stickers are bought), we substitute \(s = 0\) into the equation \(1.5s+3.5c = 21\). We get \(3.5c=21\), then \(c=\frac{21}{3.5}=6\). So, the vertical intercept \((0,6)\) means that if the teacher buys no stickers (\(s = 0\)), the teacher can buy 6 packs of cardstock.

• Horizontal intercept \((14,0)\):

When \(c = 0\) (no packs of cardstock are bought), we substitute \(c = 0\) into the equation \(1.5s+3.5c = 21\). We get \(1.5s=21\), then \(s=\frac{21}{1.5}=14\). So, the horizontal intercept \((14,0)\) means that if the teacher buys no cardstock (\(c = 0\)), the teacher can buy 14 sheets of stickers.

Answer:

a. The graph is a straight - line graph. Each point \((s,c)\) on the graph satisfies the equation \(1.5s + 3.5c=21\). For example, when \(s = 14,c = 0\): \(1.5\times14+3.5\times0=21\); when \(s = 7,c = 3\): \(1.5\times7 + 3.5\times3=10.5+10.5 = 21\); when \(s=0,c = 6\): \(1.5\times0+3.5\times6=21\).
b. The vertical intercept \((0,6)\) means that if the teacher buys no stickers (\(s = 0\)), the teacher can buy 6 packs of cardstock. The horizontal intercept \((14,0)\) means that if the teacher buys no cardstock (\(c = 0\)), the teacher can buy 14 sheets of stickers.