The table below shows several values of x and their corresponding values of y for a linear… — SAT

The table below shows several values of $x$ and their corresponding values of $y$ for a linear relationship, where $p$ and $k$ are constants.

$x$	$y$
$p$	$4$
$p + 2$	$10$
$2p + 3$	$k$

If $k = 25$ , what is the value of $p$ ?

A
0.5
4Answer
C
6
D
60

Answer

The correct answer is 4. A linear relationship has a constant rate of change (slope). Calculating the slope

m

from the first two points

(p, 4)

and

(p + 2, 10)

gives

m = \frac{10 - 4}{(p + 2) - p} = 3

. The equation of the line is

y - 4 = 3(x - p)

, which simplifies to

y = 3x - 3p + 4

. Substituting the third point

(2p + 3, 25)

into this equation gives

25 = 3(2p + 3) - 3p + 4

, which simplifies to

25 = 3p + 13

. Solving for

p

yields

p = 4

Step-by-Step Solution

Determine the slope of the linear relationship using the first two points from the table,

(p, 4)

and

(p + 2, 10)

The slope is

m = 3

A linear relationship has a constant rate of change (slope), which is the change in

y

divided by the change in

x

m = \frac{10 - 4}{(p + 2) - p} = \frac{6}{2} = 3

Write the equation of the line using the point-slope form with the point

(p, 4)

and slope

m = 3

y = 3x - 3p + 4

Expressing the linear relationship as an equation allows us to find the relationship between the parameters

p

and

k

y - 4 = 3(x - p) \Rightarrow y = 3x - 3p + 4

Substitute the third point

(2p + 3, k)

into the linear equation and simplify the expression to solve for

k

in terms of

p

k = 3p + 13

Since the point

(2p + 3, k)

lies on the line, its coordinates must satisfy the line's equation:

k = 3(2p + 3) - 3p + 4 \Rightarrow k = 6p + 9 - 3p + 4 \Rightarrow k = 3p + 13

Substitute the given value

k = 25

into the equation

k = 3p + 13

and solve for

p

p = 4

This isolates the variable

p

to find its numerical value under the specified conditions:

25 = 3p + 13 \Rightarrow 12 = 3p \Rightarrow p = 4

Key Concept

Finding the equation of a line from a table of values and using substitution to solve for unknown parameters in a linear relationship.

Alternative Method

An alternative method is to use the property that the slope between any two points on a line is constant. We can equate the slope between

(p, 4)

and

(p + 2, 10)

to the slope between

(p, 4)

and

(2p + 3, 25)

. This gives the equation

\frac{10 - 4}{(p + 2) - p} = \frac{25 - 4}{(2p + 3) - p}

, which simplifies to

3 = \frac{21}{p + 3}

. Solving this equation yields

3(p + 3) = 21

, so

p + 3 = 7

, which gives

p = 4

Estimated Time:2m 0s

Question

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Alternative Method

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