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9.02 Solutions to equations

Lesson

Introduction

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.

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

Example 1

By substituting the proposed solution into the equation, identify whether the following statements are true or false.

a

x=48 is a solution for the equation x-30=21.

Worked Solution
Create a strategy

We should replace the variable with the given value.

Apply the idea
\displaystyle 48-30\displaystyle =\displaystyle 21Substitute x=48
\displaystyle 18\displaystyle \neq\displaystyle 21Evaluate the left side and compare

Since the value of the left-hand side of the equation is not equal to the value of the right-hand side, x=48 is not a solution of the equation. The statement is false.

b

x=32 is a solution to the equation x-18=14

Worked Solution
Create a strategy

We should replace the variable with the given value.

Apply the idea
\displaystyle 32-18\displaystyle =\displaystyle 14Substitute x
\displaystyle 14\displaystyle =\displaystyle 14Evaluate

Since the value of the left-hand side of the equation is equal to the value of the right-hand side, x=32 is a solution of the equation. The statement is true.

Idea summary

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

We can use substitution to determine whether a particular value is a solution to an equation. If both sides of the equation are equal then the value is a solution.

Guess and check

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.

y12345
34+y3536373839

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:

y12345
34-y3332313029

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:

A flow diagram showing how to improve next guess. Ask your teacher for more information.

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

Example 2

Consider the equation 56-t=39.

a

Isabelle guesses that t=10 is a solution to this equation. Is she correct?

Worked Solution
Create a strategy

We can check if the value is a solution by replacing the variable with it.

Apply the idea
\displaystyle 56-10\displaystyle =\displaystyle 39Substitute t
\displaystyle 46\displaystyle \neq\displaystyle 39Evaluate

t=10 is not a solution to the equation. Isabelle's guess is not correct.

b

When substituting t=10, which side of the equation is bigger?

A
56-t
B
39
Worked Solution
Apply the idea

From part (a):

\displaystyle 56-t\displaystyle =\displaystyle 46
\displaystyle >\displaystyle 39

Option A, 56-t is bigger.

c

How can Isabelle improve her guess for the solution to the equation?

A
Guess a random number.
B
Guess a number smaller than 10.
C
Guess a number larger than 10.
D
Guess t=10 again.
Worked Solution
Create a strategy

An increase in the value of t will result in a decrease of the value of 56-t since there is a minus sign before t.

Apply the idea

Since the value of 56-t is larger than 39, Isabelle needs to decrease its value by increasing the value of t. So she should guess a number larger than 10, option C.

d

Isabelle increases her guess to t=20. When substituting this into the equation she finds that 56-t is now smaller than 39. What does this tell her about the solution to the equation?

A
The solution to the equation lies between 10 and 20.
B
The solution to the equation does not exist.
C
The solution to the equation is larger than 20.
D
The solution to the equation is smaller than 10.
Worked Solution
Apply the idea

Since the 39 lies between the values of 56-t when t=10 and when t=20, we can say that the solution to the equation lies between 10 and 20, Option A.

Use 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 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.

y10111213141516171819
4y40 60

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

y16171819
4y64687278

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

Example 3

Consider the equation t+19=35.

a

What are the values for the left-hand side and right-hand side of the equation if Danielle substitutes in t=20?

Worked Solution
Create a strategy

Replace the variable with the given value.

Apply the idea
\displaystyle \text{Left-hand side}\displaystyle =\displaystyle t+19
\displaystyle =\displaystyle 20+19Substitute t
\displaystyle =\displaystyle 39Evaluate
\displaystyle \text{Right-hand side}\displaystyle =\displaystyle 35
b

What are the values for the left-hand side and right-hand side of the equation if Danielle substitutes in t=15?

Worked Solution
Create a strategy

Replace the variable with the given value.

Apply the idea
\displaystyle \text{Left-hand side}\displaystyle =\displaystyle t+19
\displaystyle =\displaystyle 15+19Substitute t
\displaystyle =\displaystyle 34Evaluate
\displaystyle \text{Right-hand side}\displaystyle =\displaystyle 35
c

Since Danielle knows that the solution is between 15 and 20, she decides to find the solution using a table of values. Complete the table.

t16171819
t+19
Worked Solution
Create a strategy

Substitute the values from the table into the expression.

Apply the idea
\displaystyle t+19\displaystyle =\displaystyle 16+19Substitute t=16
\displaystyle =\displaystyle 35Evaluate
\displaystyle t+19\displaystyle =\displaystyle 17+19Substitute t=17
\displaystyle =\displaystyle 36Evaluate
\displaystyle t+19\displaystyle =\displaystyle 18+19Substitute t=18
\displaystyle =\displaystyle 37Evaluate
\displaystyle t+19\displaystyle =\displaystyle 19+19Substitute t=19
\displaystyle =\displaystyle 38Evaluate
t16171819
t+1935363738
d

Using the table of values from part (c), what value of t will make the equation t+19=35 true?

Worked Solution
Create a strategy

The solution to the equation will be the value for t which gives a value of 35 for t+19.

Apply the idea

From the table: t=16.

Idea summary

We can use a table of values to find the solution of an equation if the range of possible solutions is small enough.

Outcomes

MA4-10NA

uses algebraic techniques to solve simple linear and quadratic equations

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