The graph of the equation ax + by = 24, where a and b are constants, is a line in the xy-p… — SAT

The graph of the equation $ax + by = 24$ , where $a$ and $b$ are constants, is a line in the $xy$ -plane. If this line passes through the points $(2, 9)$ and $(6, 3)$ , what is the value of $a + b$ ?

Answer: 5

Answer

The value of

a + b

5

Substituting the coordinates of the two points

(2, 9)

and

(6, 3)

into the given linear equation

ax + by = 24

yields a system of two equations:

2a + 9b = 24

and

6a + 3b = 24

. Simplifying the second equation gives

2a + b = 8

, which means

b = 8 - 2a

. Substituting this expression for

b

into the first equation yields

2a + 9(8 - 2a) = 24

, which simplifies to

-16a = -48

, or

a = 3

. Plugging

a = 3

back into

b = 8 - 2a

gives

b = 2

. Thus, the value of

a + b

3 + 2 = 5

Step-by-Step Solution

Substitute the point

(2, 9)

into the equation

ax + by = 24

2a + 9b = 24

Since the line passes through the point

(2, 9)

, the coordinates must satisfy the equation of the line.

Substitute the point

(6, 3)

into the equation

ax + by = 24

6a + 3b = 24

Since the line passes through the point

(6, 3)

, the coordinates must satisfy the equation of the line.

Solve the system of equations for

a

and

b

a = 3

and

b = 2

To find the values of the constants

a

and

b

, we solve the linear system: (1)

2a + 9b = 24

and (2)

6a + 3b = 24

. Dividing the second equation by 3 gives

2a + b = 8

, or

b = 8 - 2a

. Substituting this into the first equation gives

2a + 9(8 - 2a) = 24 \implies 2a + 72 - 18a = 24 \implies -16a = -48 \implies a = 3

. Then,

b = 8 - 2(3) = 2

Add the values of

a

and

b

a + b = 5

The question asks for the value of

a + b

Key Concept

Solving systems of linear equations derived from coordinate substitution in a two-variable linear equation.

Estimated Time:1m 30s

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