In the standard (x, y) coordinate plane, a line passes through the point (3, 1) and has a … — ACT

In the standard $(x, y)$ coordinate plane, a line passes through the point $(3, 1)$ and has a slope of $\frac{2}{5}$ . If the line intersects the $y$ -axis at the point $(0, b)$ , what is the value of $b$ ?

A
$-\frac{13}{2}$
B
$-\frac{4}{5}$
C
$\frac{11}{5}$
$-\frac{1}{5}$ Cevap
E
$\frac{5}{4}$

Cevap

The value of the

y

-coordinate of the

y

-intercept is

-\frac{1}{5}

To find the

y

-intercept of the line, we can use the point-slope equation of a line:

y - y_1 = m(x - x_1)

, where

m

is the slope and

(x_1, y_1)

is a point on the line. Substituting the given point

(3, 1)

and slope

\frac{2}{5}

yields

y - 1 = \frac{2}{5}(x - 3)

. Since the

y

-intercept occurs at

x = 0

, we substitute

0

for

x

to find the corresponding

y

-value:

y - 1 = \frac{2}{5}(0 - 3)

, which simplifies to

y - 1 = -\frac{6}{5}

. Adding

1

to both sides gives

y = -\frac{1}{5}

. Therefore, the value of the

y

-coordinate of the

y

-intercept is

-\frac{1}{5}

Adım Adım Çözüm

Identify the point-slope form of a linear equation.

y - y_1 = m(x - x_1)

This formula allows us to write the equation of a line when we know its slope (

m

) and a point on the line (

(x_1, y_1)

Substitute the given slope

m = \frac{2}{5}

and point

(3, 1)

into the formula.

y - 1 = \frac{2}{5}(x - 3)

This establishes the equation for the specific line described in the problem.

Substitute

x = 0

to find the

y

-intercept of the line.

y - 1 = \frac{2}{5}(0 - 3) \implies y - 1 = -\frac{6}{5}

The

y

-intercept of any graph is the point where it crosses the

y

-axis, which always has an

x

-coordinate of

0

Solve for

y

to determine the value of

b

y = 1 - \frac{6}{5} = -\frac{1}{5}

, so

b = -\frac{1}{5}

Isolating

y

gives the

y

-coordinate of the

y

-intercept, which is

b

since the point is defined as

(0, b)

Anahtar Kavram

Writing and evaluating linear equations in point-slope form to find intercepts.

Soru

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Konular

Bu konudan daha fazla soru