8.02 Solutions to equations

Whenever we have an equation, we want to find the solution. And in order to find the solution, we want to find a value that makes the equation true.

 

Substitution

When testing which values could be the solution to an equation, we often want to use substitution to check if those values make the equation true.

We can substitute potential solutions into an equation by replacing the variable in the equation with a numeric value. After substituting the potential solution, we can evaluate both sides of the equation to check if the value we substituted is a solution. If both sides of the equation evaluate to give the same value, we have found the solution. If not, we need to try a different substitution.

 

Understanding the expression

Before we test any substitutions, however, we should first look at how our guess will affect the expression that we are substituting it into.

Consider the expression $34+y$34+y.

We can substitute some values into this expression to understand how our guesses will affect it.

$y$y	$1$1	$2$2	$3$3	$4$4	$5$5
$34+y$34+y	$35$35	$36$36	$37$37	$38$38	$39$39

We can see from the table that, as the value for $y$y increases, the value of the expression also increases. This is because $y$y is being added to $34$34. This means that if we want a greater value for the expression, we should guess a greater number.

What about the expression $34-y$34−y?

Substituting values into this expression, we get:

$y$y	$1$1	$2$2	$3$3	$4$4	$5$5
$34-y$34−y	$33$33	$32$32	$31$31	$30$30	$29$29

We can see from this table that, as $y$y increases, the value of the expression actually decreases. This is because $y$y is being subtracted from $34$34. This means that if we want a greater value for the expression, we should guess a smaller number.

Using this method, we can also see that expressions like $4y$4y and $y-3$y−3 will increase when $y$y increases, while expressions like $\frac{60}{y}$60y​ will decreases when $y$y increases.

If we are ever unsure about how a expression changes, we can just fill in a table of values!

 

Improving our guess

The substitution that we test first will not always be correct, and in those cases we simply need to try again. But how do we know what to guess next?

We can improve our guess by trying to make the expression we are changing closer to the target value. When solving the equation below, we want to make the expression on the left-hand side equal to the target value on the right-hand side.

Consider the following problem: What is the solution to the equation $4y=68$4y=68?

Let's try $y=10$y=10.

Substituting this into the left-hand side of the equation, we get:

Left-hand side = $4\times10$4×10 = $40$40

Comparing this to the right-hand side of the equation, $68$68, we can see that our value for the expression $4y$4y needs to be increased.

From our understanding of expressions, we know that the expression $4y$4y will increase when $y$y increases. This tells us that the solution to the equation will be a value for $y$y that is greater than $10$10.

 

Finding a range for the solution

Although the first guess we made was not correct, it did tell us that the solution to the equation $4y=68$4y=68 will be greater than $10$10.

Knowing this, let's try $y=20$y=20 next.

Substituting this into the left-hand side of the equation, we now get:

Left-hand side = $4\times20$4×20 = $80$80

Comparing this to the right-hand side value of $68$68, we can see that we now need to decrease the value of our expression. This tells us that the solution to the equation must be less than $20$20.

From our guesses, we know that the solution will be a value for $y$y that is greater than $10$10 and less than $20$20.

This means that the solution will be between $10$10 and $20$20.

Notice that, for our two guesses $y=10$y=10 and $y=20$y=20, the left-hand side values were $40$40 and $80$80. Another way to see that the solution to the equation is between $10$10 and $20$20 is by noticing that $68$68 lies between $40$40 and $80$80, and as $68$68 is closer to $80$80, our $y$y-value is probably closer to $20$20 than $10$10.

 

Using a table to test values

After finding a small enough range of values for the solution, we then want to test each value in the range to find the solution. We can do this by substituting each value into the equation until the equation is true.

However, this is still a fair bit of work. To save some effort, we can instead use a table of values.

As $68$68 is closer to $80$80 than $40$40, let's first test $y=15$y=15, and if $4y$4y is still less than $68$68 we know the range must be between $15$15 and $20$20. That is, we have further refined our range. We can see that $4\times15=60$4×15=60.
 

$y$y	$10$10	$11$11	$12$12	$13$13	$14$14	$15$15	$16$16	$17$17	$18$18	$19$19	$20$20
$4y$4y	$40$40					$60$60					$80$80

We have now shrunk the range of values we have to test from $16$16 to $19$19. 

$y$y	$16$16	$17$17	$18$18	$19$19
$4y$4y	$64$64	$68$68	$72$72	$78$78

We know that the solution to the equation will make the left-hand side equal in value to the right-hand side when substituted into the equation. Using the table of values, we can see that the left-hand side will be equal to $68$68 when $y=17$y=17.

Therefore, $y=17$y=17 is the solution to the equation $4y=68$4y=68.