Question

x | y\
10 | 27\
11 | a\
12 | 11\
in order for the data in the table to represent a linear function with a rate of change of -8, what must be the value of a?\
\\(\circ\\) \\(a = 2\\)\
\\(\circ\\) \\(a = 3\\)\
\\(\circ\\) \\(a = 19\\)\
\\(\circ\\) \\(a = 35\\)

Explanation:

Step1: Recall rate of change formula

The rate of change (slope) \( m \) 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 \(-8\), take points \((10,27)\) and \((11,a)\). So \( -8=\frac{a - 27}{11 - 10} \).

Step2: Solve for \( a \)

Simplify the denominator: \( 11 - 10 = 1 \). The equation becomes \( -8=\frac{a - 27}{1} \), which simplifies to \( a - 27=-8 \). Add 27 to both sides: \( a=-8 + 27=19 \). We can also check with \((11,a)\) and \((12,11)\): \( -8=\frac{11 - a}{12 - 11}=\frac{11 - a}{1} \), so \( 11 - a=-8 \), \( a = 11 + 8 = 19 \).

Answer:

a = 19