In the standard (x, y) coordinate plane, a line L passes through the point (2, 3) and has … — ACT

In the standard $(x, y)$ coordinate plane, a line $L$ passes through the point $(2, 3)$ and has a positive $y$ -intercept $(0, b)$ . The area of the triangular region in the first quadrant bounded by line $L$ , the $x$ -axis, and the $y$ -axis is $12$ square units. What is the value of $b$ ?

A
$3$
B
$4$
$6$ Cevap
D
$8$
E
$12$

Cevap

The correct value of

b

6

To find the value of

b

, we can use the intercept form of a linear equation:

\frac{x}{a} + \frac{y}{b} = 1

, where

a

is the

x

-intercept and

b

is the

y

-intercept. The area of the right triangle formed by the axes and the line is given by

\frac{1}{2}ab = 12

, which means

ab = 24

, or

a = \frac{24}{b}

. Since the line passes through the point

(2, 3)

, we substitute these coordinates into the intercept equation to get

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

. Substituting

a = \frac{24}{b}

into this equation gives

\frac{2b}{24} + \frac{3}{b} = 1

, which simplifies to

\frac{b}{12} + \frac{3}{b} = 1

. Multiplying the entire equation by

12b

to clear the denominators results in

b^2 + 36 = 12b

. Rearranging this quadratic equation gives

b^2 - 12b + 36 = 0

, which factors as

(b - 6)^2 = 0

. Solving for

b

yields

b = 6

Adım Adım Çözüm

Express the area of the right triangle in the first quadrant in terms of the

x

-intercept

(a, 0)

and

y

-intercept

(0, b)

The area is

\frac{1}{2}ab = 12

, which simplifies to

ab = 24

, or

a = \frac{24}{b}

The triangular region is a right triangle with base

a

and height

b

along the coordinate axes.

Set up the equation of the line using the intercept form

\frac{x}{a} + \frac{y}{b} = 1

and substitute the given point

(2, 3)

Substituting

x = 2

and

y = 3

gives the equation

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

Since the line passes through the point

(2, 3)

, this point must satisfy the equation of the line.

Substitute

a = \frac{24}{b}

into the intercept equation and solve the resulting quadratic equation for

b

Substituting

a

gives

\frac{2}{\frac{24}{b}} + \frac{3}{b} = 1 \implies \frac{b}{12} + \frac{3}{b} = 1

. Multiplying by

12b

yields

b^2 + 36 = 12b \implies b^2 - 12b + 36 = 0 \implies (b - 6)^2 = 0 \implies b = 6

Solving the quadratic equation yields the value of the

y

-intercept

b

Anahtar Kavram

Linear equations and graphing using intercept form and triangle area relations

Alternatif Yöntem

An alternative approach is to use the slope formula. The line passes through

(a, 0)

(2, 3)

, and

(0, b)

. The slope between

(0, b)

and

(2, 3)

\frac{3 - b}{2}

, and the slope between

(0, b)

and

(a, 0)

-\frac{b}{a}

. Equating these gives

\frac{3 - b}{2} = -\frac{b}{a} \implies 3a - ab = -2b

. Since the area is

12

, we know

ab = 24

. Substituting

ab = 24

gives

3a - 24 = -2b \implies 3a + 2b = 24

. Since

a = \frac{24}{b}

, we get

3\left(\frac{24}{b}\right) + 2b = 24 \implies \frac{72}{b} + 2b = 24 \implies b^2 - 12b + 36 = 0 \implies b = 6

Tahmini Süre:1m 30s

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