In the standard (x, y) coordinate plane, a line passes through the points (\frac{1}{2}, \f… — ACT

In the standard $(x, y)$ coordinate plane, a line passes through the points $(\frac{1}{2}, \frac{2}{3})$ and $(\frac{3}{4}, \frac{5}{6})$ . What is the $y$ -intercept of this line?

$\frac{1}{3}$ Answer
B
$\frac{1}{6}$
C
$-1$
D
$-\frac{1}{12}$
E
$-\frac{1}{7}$

Answer

The

y

-intercept of the line is

\frac{1}{3}

The correct answer is

\frac{1}{3}

. By calculating the slope of the line as

m = \frac{2}{3}

, and using the point-slope formula

y - y_1 = m(x - x_1)

with either given point, we can isolate

y

to get the equation

y = \frac{2}{3}x + \frac{1}{3}

, where the constant term represents the

y

-intercept.

Step-by-Step Solution

Calculate the slope (

m

) of the line passing through the points

(\frac{1}{2}, \frac{2}{3})

and

(\frac{3}{4}, \frac{5}{6})

using the slope formula

m = \frac{y_2 - y_1}{x_2 - x_1}

m = \frac{\frac{5}{6} - \frac{2}{3}}{\frac{3}{4} - \frac{1}{2}} = \frac{\frac{1}{6}}{\frac{1}{4}} = \frac{2}{3}

To find the equation of a line, we first need to determine its slope.

Use the point-slope form

y - y_1 = m(x - x_1)

with the point

(\frac{1}{2}, \frac{2}{3})

and slope

m = \frac{2}{3}

to write the equation of the line.

y - \frac{2}{3} = \frac{2}{3}(x - \frac{1}{2}) \Rightarrow y - \frac{2}{3} = \frac{2}{3}x - \frac{1}{3}

This allows us to write the linear equation in a form where we can isolate the

y

-intercept.

Isolate

y

to convert the equation into slope-intercept form

y = mx + b

, where

b

represents the

y

-intercept.

y = \frac{2}{3}x + \frac{1}{3}

The constant term in slope-intercept form represents the

y

-intercept.

Key Concept

Linear Equations and Graphing

Alternative Method

Instead of using the point-slope form, you can substitute one of the points directly into the slope-intercept form

y = mx + b

. Substituting

(\frac{1}{2}, \frac{2}{3})

and

m = \frac{2}{3}

yields

\frac{2}{3} = \frac{2}{3}(\frac{1}{2}) + b

, which simplifies to

\frac{2}{3} = \frac{1}{3} + b

. Solving for

b

gives

b = \frac{1}{3}

Estimated Time:1m 30s

Question

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Alternative Method

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