In the standard (x, y) coordinate plane, a line with a negative slope passes through the p… — ACT

In the standard $(x, y)$ coordinate plane, a line with a negative slope passes through the point $(4, 3)$ and has an $x$ -intercept that is twice its $y$ -intercept. If the equation of this line is written in the form $Ax + By = C$ , where $A$ , $B$ , and $C$ are integers with no common factor greater than 1, and $A > 0$ , what is the value of $C$ ?

Answer: 10

Answer

The value of

C

is 10.

The correct answer is 10. By setting the intercepts as

(2b, 0)

and

(0, b)

, we find the slope is

m = -\frac{1}{2}

. Applying the point-slope formula with

(4, 3)

gives

y - 3 = -\frac{1}{2}(x - 4)

, which simplifies to

y = -\frac{1}{2}x + 5

. Converting this to standard form with

A > 0

yields

x + 2y = 10

, where

C = 10

Step-by-Step Solution

Represent the coordinates of the intercepts using a variable.

The

y

-intercept is

(0, b)

and the

x

-intercept is

(2b, 0)

The problem states that the

x

-intercept is twice the

y

-intercept.

Calculate the slope (

m

) of the line using the intercepts.

m = \frac{b - 0}{0 - 2b} = -\frac{1}{2}

The slope of a line passing through

(x_1, y_1)

and

(x_2, y_2)

is given by

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

Find the equation of the line using the point-slope form with point

(4, 3)

y - 3 = -\frac{1}{2}(x - 4) \Rightarrow y = -\frac{1}{2}x + 5

The point-slope form of a linear equation is

y - y_1 = m(x - x_1)

Convert the equation to the standard form

Ax + By = C

x + 2y = 10

Multiplying the equation by 2 and moving the

x

term to the left side results in integer coefficients where the coefficient of

x

is positive (

A = 1 > 0

Identify the value of

C

from the standard form equation.

C = 10

Comparing

x + 2y = 10

with

Ax + By = C

shows that

A=1

B=2

, and

C=10

. These integers share no common factors greater than 1.

Key Concept

Converting a linear equation to standard form using given geometric features and a coordinate point.

Alternative Method

Instead of using the point-slope formula, substitute the point

(4, 3)

directly into the intercept form of a linear equation,

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

. Since

a = 2b

, this becomes

\frac{4}{2b} + \frac{3}{b} = 1

. Simplifying this gives

\frac{2}{b} + \frac{3}{b} = 1 \Rightarrow \frac{5}{b} = 1 \Rightarrow b = 5

. Thus,

a = 10

. The intercept form is

\frac{x}{10} + \frac{y}{5} = 1

. Multiplying by 10 to clear denominators gives

x + 2y = 10

, where

C = 10

Estimated Time:1m 30s

Question

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Alternative Method

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