x | y\n3 | 13\n4 | a\n5 | 23\nin order for the data in the table to represent a linear function with a rate…
x | y\n3 | 13\n4 | a\n5 | 23\nin 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?\n○ a = 3\n○ a = 8\n○ a = 18\n○ a = 33
Answer
Explanation:
Step1: Recall rate of change formula
The rate of change (slope) ( m ) of a linear function between two points ((x_1,y_1)) and ((x_2,y_2)) is ( m=\frac{y_2 - y_1}{x_2 - x_1} ). Here, rate of change is ( +5 ), and we can use points ((3,13)) and ((4,a)), or ((4,a)) and ((5,23)). Let's use ((3,13)) and ((4,a)): ( 5=\frac{a - 13}{4 - 3} ).
Step2: Solve for ( a )
Simplify the denominator: ( 4 - 3 = 1 ). So the equation becomes ( 5=\frac{a - 13}{1} ), which simplifies to ( a - 13 = 5 ). Add 13 to both sides: ( a = 5 + 13 = 18 ). We can check with the other pair ((4,a)) and ((5,23)): ( 5=\frac{23 - a}{5 - 4}\Rightarrow 5=\frac{23 - a}{1}\Rightarrow 23 - a = 5\Rightarrow a = 23 - 5 = 18 ), which matches.
Answer:
( a = 18 ) (corresponding to the option "a = 18")