In the system of equations below, a and b are constants. \begin{aligned} ax + by &= 24 \\ … — SAT

In the system of equations below, $a$ and $b$ are constants.

\begin{aligned} ax + by &= 24 \\ 2x - 5y &= -7 \end{aligned}

If the system has the same unique solution $(x, y)$ for all values of $a$ and $b$ that satisfy the equation $4a + 3b = 24$ , what is the value of $x + y$ ?

Answer: 7

Answer

The value of

x + y

7

The correct answer is

7

. The solution to the system must satisfy

ax + by = 24

for any constants

a

and

b

that satisfy the constraint

4a + 3b = 24

. By matching the coefficients of

a

and

b

in both equations, we find

x = 4

and

y = 3

. To verify, we substitute these coordinates into the second equation:

2(4) - 5(3) = 8 - 15 = -7

, which is correct. The sum of the coordinates is

x + y = 4 + 3 = 7

Step-by-Step Solution

Analyze the constraint on the constants

a

and

b

given by

4a + 3b = 24

and relate it to the first equation

ax + by = 24

Since the first equation

ax + by = 24

must hold for all pairs of

(a, b)

satisfying

4a + 3b = 24

, the coefficients must correspond directly, meaning

x = 4

and

y = 3

This shows that

(4, 3)

is a point that lies on the line

ax + by = 24

regardless of the specific values of

a

and

b

, as long as they satisfy the given constraint.

Prove the uniqueness of the point

(4, 3)

by testing specific values for

(a, b)

that satisfy the constraint

4a + 3b = 24

b = 0

, then

4a = 24 \implies a = 6

. The equation becomes

6x = 24 \implies x = 4

. If

a = 0

, then

3b = 24 \implies b = 8

. The equation becomes

8y = 24 \implies y = 3

. This confirms

(x, y) = (4, 3)

is the unique point.

Choosing convenient test values for

a

and

b

helps isolate the variables

x

and

y

to verify the coordinates of the solution.

Substitute the point

(4, 3)

into the second equation of the system,

2x - 5y = -7

, to verify that it is consistent.

2(4) - 5(3) = 8 - 15 = -7

. Since this statement is true,

(4, 3)

is the consistent unique solution to the system of equations.

A solution to a system of equations must satisfy all equations in the system.

Find the requested value of

x + y

using the coordinates of the solution.

x + y = 4 + 3 = 7

The question asks for the sum of the coordinates of the solution.

Key Concept

Systems of linear equations with parameter constraints

Question

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Step-by-Step Solution

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