System of equations help

Question

How do you get $a=2$ and $d=5$ from the two equations (see where I marked it)? Thank you!

Answer 1

We wish to solve the system of equations:

$$\begin{cases} 52=a+10d~~~~~~~(*)\\92=a+18d~~~~~~~(\dagger)\end{cases}$$

This can be done a number of ways. One of the earliest methods learned is that of substitution or elimination.

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By Elimination:

By taking equation $(\dagger)$ and subtracting equation $(*)$, we get:

$\begin{array}{lrl} &92&=a+18d\\-~~~&52&=a+10d\\\hline &40&=~~~~~~~~8d\end{array}$

Dividing by $8$ you get that $d=5$. Taking this information and putting it back into either of the equations, (I'll use $(*)$) you get:

$52 = a + 10d = a + 10(5) = a+50$

$2 = a$

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By Substitution:

We wish to isolate one of the variables. It doesn't matter which and it doesn't matter which equation we do it from. It appears convenient to do so using $(*)$ and $a$.

$\begin{array}{rlr}52&=a+10d\\ 52-10d&=a&~~~~~~~(\heartsuit)\end{array}$

Now we have a way of expressing $a$ in terms of things that are not $a$. Let us use this new way of expressing $a$ in the equation that we did not use. (note: if you use this into the equation you used to derive the relationship, you will just arrive at $0=0$ which is uninteresting)

$\begin{array}{rl} 92&=a+18d\\ 92&=(52-10d)+18d\\92&=52+8d\\40&=8d\\5&=d\end{array}$

We then use this knowledge of $d$ into $(\heartsuit)$ to get:

$a = 52-10d = 52 - 10(5) = 52-50=2$

Answer 2

Each of your two expressions can be reconfigured to give an equation for $a$ so that $$a=92-18d=52-10d$$This enables you to eliminate $a$, and obtain an equation involving only $d$ $$92-18d=52-10d$$which is solved in the usual way to obtain $d=5$. Then either of the expressions for $a$ can be used to compute $a=2$.

When you get used to this kind of equation, it is very easy to solve. There are several possible approaches, all of which are useful. Circumstances may make one easier than another. Some methods are applicable to more general problems, like three linear equations involving three unknown quantities.