Question

the tables represent two linear functions in a system. what is the solution to this system? (1,0) (1,6) (8,26) (8, - 22)

Explanation:

Step1: Recall solution of system of linear - functions

The solution of a system of two linear functions is the point \((x,y)\) that satisfies both functions. This is the point of intersection of the two lines represented by the functions. We can find the \(x\) - value where the \(y\) - values in both tables are the same.

Step2: Compare \(y\) - values in the tables

In the first table, when \(x = 2\), \(y = 2\); when \(x=0\), \(y = 10\); when \(x = 2\), \(y = 18\); when \(x = 4\), \(y = 26\). In the second table, when \(x = 2\), \(y=-4\); when \(x = 0\), \(y = 2\); when \(x = 2\), \(y = 8\); when \(x = 4\), \(y = 14\).
We can also assume the linear - function form \(y=mx + b\) for each table. For the first table, using the two - point form \(m=\frac{y_2 - y_1}{x_2 - x_1}\). Let \((x_1,y_1)=(0,10)\) and \((x_2,y_2)=(2,18)\), then \(m=\frac{18 - 10}{2-0}=4\), and \(b = 10\) (since when \(x = 0,y = 10\)), so \(y_1=4x + 10\). For the second table, let \((x_1,y_1)=(0,2)\) and \((x_2,y_2)=(2,8)\), then \(m=\frac{8 - 2}{2-0}=3\), and \(b = 2\), so \(y_2=3x + 2\).
We set \(y_1=y_2\), so \(4x+10 = 3x + 2\).

Step3: Solve the equation \(4x+10 = 3x + 2\) for \(x\)

Subtract \(3x\) from both sides: \(4x-3x+10=3x - 3x+2\), which gives \(x+10 = 2\). Then subtract 10 from both sides: \(x=2 - 10=-8\).
Substitute \(x=-8\) into \(y = 3x + 2\) (we could also use the other equation), \(y=3\times(-8)+2=-24 + 2=-22\).

Answer:

\((-8,-22)\)