Final answer:
Increasing the value of x by 1 in the equation y = 8x + 2 results in y increasing by a factor of eight, as the slope (8) indicates how much y will increase for each unit increase in x.
Step-by-step explanation:
When the value of x increases by 1 in the linear equation y = 8x + 2, the value of y will also increase. However, to determine by how much y will increase, we need to recognize that the coefficient of x in the equation (which is 8) is the slope of the line. The slope indicates that for each unit increase in x, y will increase by 8 units. So, if x increases by 1, y increases by 8 times that amount, which means y will increase by a factor of eight.
For example, if you start with x = 1, then y = 8(1) + 2 = 10. If x increases to 2, then y becomes 8(2) + 2 = 18. The increase from 10 to 18 is by 8 units.
Linear equations like y = 6x + 8, 4y = 8, and y + 7 = 3x have a similar behavior where y depends on x, the independent variable. The graph of such an equation shows a straight line, and the slope (m) quantifies the increase or decrease of y relative to x.