8.02 Solutions to equations

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.

Substitution involves replacing one expression with another that is proposed to be equal to it.

We can substitute potential solutions into an equation by replacing the pronumeral 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.

Examples

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.

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

We can see from the table that, as the value for y increases, the value of the expression also increases. This is because y is being added to 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? Substituting values into this expression, we get:

We can see from this table that, as y increases, the value of the expression actually decreases. This is because y is being subtracted from 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 and y-3 will increase when y increases, while expressions like \dfrac{60}{y} will decrease when y increases. If we are ever unsure about how a expression changes, we can just fill in a table of values.

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.

We can use this flow diagram to help us improve our next guess:

Consider the following problem: What is the solution to the equation 4y=58? Let's try y=10.

Substituting this into the left-hand side of the equation, we get: \text{Left-hand side} = 4\times10 = 40

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

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

Although the first guess we made was not correct, it did tell us that the solution to the equation \\ 4y=68 will be greater than 10. Knowing this, let's try y=20 next.

Substituting this into the left-hand side of the equation, we now get: \text{Left-hand side }= 4\times20 = 80

Comparing this to the right-hand side value of 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.

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

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

Caution: Just because we have found a range does not necessarily mean that it is useful. For example: y=0 gives us a left-hand side value of 0 and y=100 gives us a left-hand side value of 400. Since 68 lies between 0 and 400, the solution must lie between 0 and 100. This is true, but not very helpful. In this case, we would like to test more values to find a smaller range.

Examples

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

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

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 when y=17.

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

Examples