Question

in order for the data in the table to represent a linear function with a rate of change of +5, what must be the value of ( a )?

( x )	( y )
4	( a )
5	23

options: ( a = 8 ); ( a = 18 ); ( a = 33 ); ( a = 3 )

Explanation:

Step1: Recall rate of change formula

For a linear function, the rate of change (slope) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is \(\frac{y_2 - y_1}{x_2 - x_1}\). Here, the rate of change is \(+5\). Let's take the points \((3, 13)\) and \((4, a)\).

Step2: Apply the rate of change formula

Using the formula, \(\frac{a - 13}{4 - 3}=5\). Since \(4 - 3 = 1\), the equation simplifies to \(a - 13 = 5\times1\).

Step3: Solve for \(a\)

Adding \(13\) to both sides of the equation \(a - 13 = 5\), we get \(a = 5 + 13\).

Step4: Calculate the value of \(a\)

\(5+13 = 18\), so \(a = 18\).

Answer:

\(a = 18\) (corresponding to the option: \(a = 18\))