In the xy-plane, the system of linear equations below has infinitely many solutions, where… — SAT

In the $xy$ -plane, the system of linear equations below has infinitely many solutions, where $a$ , $b$ , and $c$ are constants and $c > 0$ :

\begin{aligned} 3x - 4y &= 10 \\ ax + by &= c \end{aligned}

If the graph of the second equation in the system passes through the point $(a, b)$ , what is the value of $c$ ?

Answer: 4

Answer

The correct answer is 4.

Since the system has infinitely many solutions, the two equations are equivalent. This means the coefficients are proportional, so we can write

a = 3k

b = -4k

, and

c = 10k

for some constant

k

. Because the line passes through

(a, b)

, substituting these coordinates into the second equation gives

a^2 + b^2 = c

. Substituting the expressions in terms of

k

results in

(3k)^2 + (-4k)^2 = 10k

, which simplifies to

25k^2 = 10k

. Since

c > 0

, we have

k > 0

, and dividing by

k

gives

25k = 10

, so

k = 0.4

. Finally,

c = 10(0.4) = 4

Step-by-Step Solution

Set up a proportionality constant to relate the coefficients of the two equations.

a = 3k

b = -4k

, and

c = 10k

for a constant

k

Since the system has infinitely many solutions, the equations represent the same line, meaning their coefficients and constants must be proportional.

Substitute the point

(a, b)

into the second equation

ax + by = c

a^2 + b^2 = c

The graph of the equation passes through the point

(a, b)

, so the coordinates must satisfy the equation.

Substitute the parametric expressions of

a

b

, and

c

into the equation

a^2 + b^2 = c

25k^2 = 10k

This allows us to solve for the parameter

k

using a single variable quadratic equation.

Solve the equation

25k^2 = 10k

for

k

, given that

k > 0

k = 0.4

Since

c > 0

and

c = 10k

k

must be strictly positive, allowing us to divide both sides by

k

Compute the final value of

c

using the value of

k

c = 4

Substituting

k = 0.4

back into the expression

c = 10k

gives the value of

c

Key Concept

Systems of linear equations with infinitely many solutions and coordinate geometry constraints

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