Questions: Select your answer Which equation represents a line that contains the points (2,3) and (3,5) ? y=3 x +4 y=x +1 y=5 x - 10 y=2 x - 1

Transcript text: Select your answer Which equation represents a line that contains the points $(2,3)$ and $(3,5)$ ? \[ \begin{array}{l} y=3 x \\ +4 \end{array} \] \[ \begin{array}{l} y=x \\ +1 \end{array} \] $y=5 x-$ 10 $y=2 x$ - 1

Solution

Solution Steps

To find the equation of a line that contains the points (2,3) and (3,5), we need to determine the slope of the line using the formula for slope between two points, $ m = \frac{y_2 - y_1}{x_2 - x_1} $. Once we have the slope, we can use the point-slope form of a line equation, $ y - y_1 = m(x - x_1) $, to find the equation of the line. Finally, we can simplify this equation to match it with one of the given options.

Step 1: Calculate the Slope

To find the slope $ m $ of the line that passes through the points $ (2, 3) $ and $ (3, 5) $, we use the formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{5 - 3}{3 - 2} = \frac{2}{1} = 2.0 \]

Step 2: Determine the Y-Intercept

Using the point-slope form of the line equation $ y - y_1 = m(x - x_1) $, we can find the y-intercept $ b $. Substituting $ m = 2.0 $ and the point $ (2, 3) $: \[ b = y_1 - m \cdot x_1 = 3 - 2.0 \cdot 2 = 3 - 4 = -1.0 \]

Step 3: Write the Equation of the Line

Now that we have both the slope and the y-intercept, we can write the equation of the line in slope-intercept form: \[ y = mx + b \implies y = 2.0x - 1.0 \]