Question

in slope-intercept form, what is the equation of the line passing through the points (2, 13) and (5, 22)?
\\( y = \frac{1}{3}x + 7 \\)
\\( y = -3x + 19 \\)
\\( y = 3x + 7 \\)
\\( y = 3x - 7 \\)

Explanation:

Step1: Calculate the slope

The slope \( m \) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by \( m=\frac{y_2 - y_1}{x_2 - x_1} \). For the points \((2, 13)\) and \((5, 22)\), we have \( x_1 = 2,y_1 = 13,x_2 = 5,y_2 = 22 \). So \( m=\frac{22 - 13}{5 - 2}=\frac{9}{3}=3 \).

Step2: Use point - slope form to find the equation

The point - slope form of a line is \( y - y_1=m(x - x_1) \). We can use the point \((2, 13)\) and \( m = 3 \). Substitute into the formula: \( y-13 = 3(x - 2) \).

Step3: Simplify the equation

Expand the right - hand side: \( y-13=3x-6 \). Then add 13 to both sides: \( y=3x - 6 + 13=3x+7 \). We can also check by substituting the points into the equation. For \( x = 2 \), \( y=3\times2 + 7=6 + 7 = 13 \) (which matches the point \((2,13)\)). For \( x = 5 \), \( y=3\times5+7 = 15 + 7=22 \) (which matches the point \((5,22)\)).

Answer:

\( y = 3x + 7 \) (the option with the equation \( y = 3x+7 \))