If $(1,1),(2,3),(k,5)$ are members of the function $\{(x,y)|y=4ax-b\}$, then $k=~?$
Determining a point of a linear equation
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algebra-precalculus
2 Answers
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I'll assume the function reads, $y=4ax-b$. Plug in the first two points to obtain a system of two equations in two variables. That is, $$1=4a-b,$$ and $$3=8a-b.$$ Then solve for $a$ and $b$ to find the function $y(x)$. You can then just plug in the point $(k,5)$ to solve for $k$, as one equation in one variable.
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I suppose the purpose of this exercise is to introduce the notion of "slope." The slope of a line or linear function is a constant that tells us how much faster (or slower) $y$ increases (or decreases) than $x$. By looking at the first two points, we see that when $x$ increases by $1$, $y$ increases by $2$. So looking at the second two points, we see $y$ increases by $2$ again, so it must be that $x$ increases by $1$ again, when moving from the second point to the third.