Question

1. the resistance of a material depends on the temperature and can be described by a linear equation. for a certain material, the resistance is given by the equation \\(r = 0.0045t + 8.54\\), where \\(t\\) is the temperature in degrees celsius.

(a) what is the slope of this line?

(b) what is the \\(y\\)-intercept?

1. challenge: graph these equations on the same coordinate plane:

\\(y = \frac{2}{3}x\\), \\(y = \frac{2}{3}x + 2\\), and \\(y = \frac{2}{3}x - 5\\).
identify the slope and the \\(y\\)-intercept of each line. what is true about a set of lines that all have the same slope?

Explanation:

Identify slope and intercept of the resistance equation

Using the Slope-Intercept Form knowledge point

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Identify slope and y-intercept for the challenge equations

Using the Slope-Intercept Form and y-intercept knowledge points

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Analyze geometric relationship of lines with equal slopes

When different linear equations share the exact same slope \(m\) but have different \(y\)-intercepts, they rise or fall at the exact same rate. Geometrically, this means the lines will never intersect. Therefore, a set of lines with the same slope are parallel to each other.

Answer:

Question 11

(a) The slope of the line is \(0.0045\).
(b) The \(y\)-intercept is \(8.54\) (or the point \((0, 8.54)\)).

Question 12

• For \(y = \frac{2}{3}x\): Slope is \(\frac{2}{3}\), \(y\)-intercept is \(0\) (or \((0, 0)\)).
• For \(y = \frac{2}{3}x + 2\): Slope is \(\frac{2}{3}\), \(y\)-intercept is \(2\) (or \((0, 2)\)).
• For \(y = \frac{2}{3}x - 5\): Slope is \(\frac{2}{3}\), \(y\)-intercept is \(-5\) (or \((0, -5)\)).

Conclusion: Lines that have the same slope are parallel to each other.