Consider the system of linear equations in two variables x and y: \begin{aligned} 2x - 3y … — GRE General Test

Consider the system of linear equations in two variables

x

and

y

\begin{aligned} 2x - 3y &= a \\ 4x - 6y &= b \end{aligned}

where

a

and

b

are real constants. Which of the following statements MUST be true? Select all that apply.

If $b = 2a$ , the system has infinitely many solutions.Answer
If $b \neq 2a$ , the system has no solution.Answer
C
If $a = 0$ and $b = 0$ , the system has a unique solution at $(0, 0)$ .
D
There exist real values of $a$ and $b$ for which the system has exactly one solution.
If $b = 2a + 1$ , the lines represented by the equations in the $xy$ -plane are parallel and distinct.Answer

Answer

The correct statements are: 'If

b = 2a

, the system has infinitely many solutions.', 'If

b \neq 2a

, the system has no solution.', and 'If

b = 2a + 1

, the lines represented by the equations in the

xy

-plane are parallel and distinct.'

Statements asserting that the system has infinitely many solutions when

b = 2a

, no solution when

b \neq 2a

, and parallel distinct lines when

b = 2a + 1

are all mathematically sound. Multiplying the first equation by 2 reveals that the left sides are identical (

4x - 6y

). Equality of the right sides (

b = 2a

) makes the lines identical, whereas inequality (

b \neq 2a

) makes them parallel and distinct.

Step-by-Step Solution

Analyze the coefficients of the system of equations.

The coefficients of

x

and

y

in the second equation (

4

and

-6

) are exactly twice the coefficients of

x

and

y

in the first equation (

2

and

-3

Comparing coefficient ratios determines line relationships (slopes).

Multiply the first equation by 2.

2(2x - 3y) = 2(a) \implies 4x - 6y = 2a

This puts the left side of the first equation in exact alignment with the second equation (

4x - 6y = b

Evaluate the condition for infinitely many solutions.

b = 2a

, the two equations become

4x - 6y = 2a

and

4x - 6y = 2a

, which describe the exact same line, giving infinitely many solutions.

Coincident lines intersect at every point along the line.

Evaluate the condition for no solution.

b \neq 2a

, subtracting the equations gives

0 = b - 2a \neq 0

, a contradiction. Hence, the lines are parallel and distinct, meaning no solution exists.

Distinct parallel lines never intersect.

Examine specific cases such as

b = 2a + 1

and

a=0, b=0

For

b = 2a + 1

, since

2a + 1 \neq 2a

b \neq 2a

, confirming parallel distinct lines. For

a=0, b=0

b = 2(0) = 0

, which yields infinitely many solutions rather than a unique solution.

Verifies specific option claims.

Key Concept

Systems of Linear Equations (Solvability and Geometric Interpretation)

Question

Answer

Step-by-Step Solution

Key Concept

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