Question

write the equation of the line that passes through the points \\((-1, 2)\\) and \\((6, 3)\\) in slope-intercept form.

step 2: \\(x_2 = 6\\) \\(y_2 = 3\\)

step 3: apply the slope formula: \\(m = \frac{y_2 - y_1}{x_2 - x_1}\\)
\\(m = \frac{3 - 2}{6 - (-1)} = \frac{1}{7}\\)

step 4: use one of the given points to find the \\(y\\)-intercept by substituting values for \\(x\\), \\(y\\), and \\(m\\) into the equation \\(y = mx + b\\).
for example, if \\(x = -1\\) and \\(y = 2\\), solve the equation \\(2 = \frac{1}{7}(-1) + b\\), \\(b = \frac{15}{7}\\)

what is the equation of the line in slope-intercept form?

\\(y = \frac{1}{7}x - \frac{15}{7}\\)
\\(y = \frac{1}{7}x + \frac{15}{7}\\)
\\(y = \frac{15}{7}x - \frac{1}{7}\\)
\\(y = \frac{15}{7}x + \frac{1}{7}\\)

Explanation:

Calculate the slope of the line

Using the Slope Formula knowledge point

$$ m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{3 - 2}{6 - (-1)} = \frac{1}{7} $$

Find the y-intercept

Using the Linear Equation Solving knowledge point

$$
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Write the final equation

Using the Slope-Intercept Form knowledge point

$$ y = \frac{1}{7}x + \frac{15}{7} $$

Analyze the incorrect options

We examine why the other choices are incorrect to help troubleshoot common algebraic errors:

• Option A: \(y = \frac{1}{7}x - \frac{15}{7}\). This option has the correct slope but uses a negative sign for the \(y\)-intercept, which occurs if one incorrectly subtracts \(\frac{1}{7}\) from \(2\) instead of adding it when solving \(2 = -\frac{1}{7} + b\).
• Option C: \(y = \frac{15}{7}x - \frac{1}{7}\). This option incorrectly swaps the calculated slope \(m = \frac{1}{7}\) and the \(y\)-intercept \(b = \frac{15}{7}\), and also has a sign error on the constant term.
• Option D: \(y = \frac{15}{7}x + \frac{1}{7}\). This option also swaps the slope and the \(y\)-intercept values.

Answer:

• (A) \(y = \frac{1}{7}x - \frac{15}{7}\)
• (B) \(y = \frac{1}{7}x + \frac{15}{7}\) (Correct answer)
• (C) \(y = \frac{15}{7}x - \frac{1}{7}\)
• (D) \(y = \frac{15}{7}x + \frac{1}{7}\)