Problem 1

Given two lines $y = 2x + 3$ and $y = ax + b$ have an intersection point. Determine the intersection point!

Decide whether statements $(1)$ and $(2)$ below are sufficient to answer the question.

$a \neq 2$
$b = 3$

Statement $(1)$ ALONE is sufficient, but statement $(2)$ alone is not sufficient.

Statement $(2)$ ALONE is sufficient, but statement $(1)$ alone is not sufficient.

BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

EACH statement ALONE is sufficient.

Statements $(1)$ and $(2)$ TOGETHER are NOT sufficient.

Explanation

We are asked to determine the intersection point of two lines:

$y = 2x + 3$
$y = ax + b$

To determine a unique intersection point $(x, y)$ , we must be able to solve this system of equations. Let's equate the two equations:

2x + 3 = ax + b

2x - ax = b - 3

(2 - a)x = b - 3

From the equation above, the value of $x$ can be determined if and only if the coefficient of $x$ is not zero, i.e., $2 - a \neq 0$ or $a \neq 2$ . If $a \neq 2$ , then:

x = \frac{b - 3}{2 - a}

After finding $x$ , we can find $y$ by substituting it into one of the equations. So, to determine the intersection point specifically, we need to know:

That $a \neq 2$ (so the lines intersect at one point).
The value of $b$ (to calculate the coordinate values).

Analysis of Statement 1

Statement $(1)$ provides the information $a \neq 2$ . This guarantees that the two lines have different gradients ( $m_1 = 2$ and $m_2 = a \neq 2$ ), so they definitely intersect at one point. However, we do not know the value of $b$ . Without the value of $b$ , we cannot determine the coordinates of the intersection point (values of $x$ and $y$ still depend on $b$ ). Therefore, statement $(1)$ alone is not sufficient.

Analysis of Statement 2

Statement $(2)$ provides the information $b = 3$ . The second equation becomes $y = ax + 3$ . We do not know the value of $a$ .

If $a = 2$ , then the line becomes $y = 2x + 3$ , which coincides with the first line (infinitely many intersection points).
If $a \neq 2$ , the lines intersect at one point.

Since $a$ is unknown, we cannot ensure there is a single intersection point, nor can we determine its coordinates. Therefore, statement $(2)$ alone is not sufficient.

Analysis of Both Statements Together

If we combine both statements:

$a \neq 2$
$b = 3$

We substitute into the equation to find $x$ :

x = \frac{3 - 3}{2 - a} = \frac{0}{2 - a} = 0

(Division by $2 - a$ is valid because $a \neq 2$ ). Then we find $y$ :

y = 2(0) + 3 = 3

We get a unique intersection point $(0, 3)$ . Since we can determine the intersection point, both statements together are sufficient.

Therefore, BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

Command Palette

Number 1

Explanation

Analysis of Statement 1

Analysis of Statement 2

Analysis of Both Statements Together