How Many Solutions? Transform, Read, and Justify | Grade 8 Math | LevelKite

You already know how to solve an equation that has the variable on both sides. This lesson asks a new question: how many numbers make an equation true? The answer is not always one.

Here are the words you need:

A variable is a letter, such as $x$ , that stands for a number. In this lesson $x$ may be any real number, which means any number on the number line, unless a problem says otherwise. These permitted values are called the domain of $x$ .
A linear equation in $x$ is an equation in which each side can be simplified to a number times $x$ plus a number, such as $2x + 1 = x + 3$ .
A solution is a permitted value of $x$ that makes the equation a true statement.

Start by testing values in three linear equations:

Equation A: $2x + 1 = x + 3$
Equation B: $x + 2 = x + 5$
Equation C: $2(x + 1) = 2x + 2$

A value of $x$ is a solution when the left side and the right side come out equal. For example, when $x = 2$ , Equation A has left side $2(2) + 1 = 5$ and right side $2 + 3 = 5$ , so $2$ is a solution of Equation A. The table shows the left side and the right side of each equation for five values of $x$ .

Left side and right side of each equation for five tested values

Tested value of x	Equation A	Equation B	Equation C
$-1$	Left $-1$ , right $2$ : not equal	Left $1$ , right $4$ : not equal	Left $0$ , right $0$ : equal
$0$	Left $1$ , right $3$ : not equal	Left $2$ , right $5$ : not equal	Left $2$ , right $2$ : equal
$1$	Left $3$ , right $4$ : not equal	Left $3$ , right $6$ : not equal	Left $4$ , right $4$ : equal
$2$	Left $5$ , right $5$ : equal	Left $4$ , right $7$ : not equal	Left $6$ , right $6$ : equal
$3$	Left $7$ , right $6$ : not equal	Left $5$ , right $8$ : not equal	Left $8$ , right $8$ : equal

Look down each column. In Equation A, only $x = 2$ makes the two sides equal. In Equation B, none of the tested values works. In Equation C, every tested value works.

That suggests three different answers to how many: one, none, and all. But a table tests only a few numbers, and $x$ may be any real number. Perhaps an untested value, such as $x = 2.5$ or $x = -40$ , behaves differently. To be sure about every value at once, you transform the equation: you rewrite it, step by step, as a simpler equation that has exactly the same solutions.

Two equations with exactly the same solutions are called equivalent equations. Two kinds of steps, called valid steps in this lesson, always turn an equation into an equivalent one:

Replace a side with an equivalent expression. Equivalent expressions have the same value for every permitted value of $x$ . Distributing and combining like terms give equivalent expressions, so they do not change which values make the equation true.
Do the same operation to both sides. You may add or subtract the same number or the same term, such as $3x$ , or multiply or divide by the same number, as long as that number is not zero.

A both-sides step keeps the solutions the same because you can undo it with the opposite operation, so no solution is gained or lost. Zero is the exception: dividing by zero is undefined, and multiplying both sides by zero turns every equation into $0 = 0$ , which you cannot undo.

Watch what valid steps do to Equations A, B and C.

The same three equations after valid steps

Original equation	Valid steps	Final equation	What the final equation says
A: $2x + 1 = x + 3$	Subtract $x$ from both sides to get $x + 1 = 3$ . Then subtract $1$ from both sides.	$x = 2$	True for exactly one value, $2$
B: $x + 2 = x + 5$	Subtract $x$ from both sides.	$2 = 5$	False, whatever $x$ is
C: $2(x + 1) = 2x + 2$	Distribute to get $2x + 2 = 2x + 2$ . Then subtract $2x$ from both sides.	$2 = 2$	True, whatever $x$ is

Each final equation has exactly the same solutions as the equation it came from. So now you know about every real number, not just the five you tested.

$x = 2$ is true for $2$ and for no other number. Equation A has exactly one solution.
$2 = 5$ has no $x$ in it, so no choice of $x$ can make it true. Equation B has no solution. That makes sense: adding $2$ to a number never gives the same result as adding $5$ to it.
$2 = 2$ has no $x$ in it, so no choice of $x$ can make it false. Every real number is a solution of Equation C. That makes sense too: $2(x + 1)$ and $2x + 2$ are equivalent expressions.

Key concept

The final equation shows the number of solutions

Valid steps keep the solutions the same. You may replace a side with an equivalent expression, or do the same operation to both sides: add or subtract the same term, or multiply or divide by the same nonzero number. Never divide by zero.

When $x$ may be any real number, valid steps bring a linear equation to one of three final forms, with $x$ alone on one side or with no $x$ left:

$x = a$ for one number $a$ : exactly one solution. Check it in the original equation.
A false statement such as $2 = 5$ : the equation is a contradiction, true for no value. No solution.
A true statement such as $2 = 2$ : the equation is an identity, true for every permitted value. Every real number is a solution: infinitely many solutions.

Worked example

Exactly one solution

How many solutions does $5x - 3 = 2x + 12$ have? Here $x$ may be any real number.

Subtract 2x from both sides so that x appears on one side only. Subtracting the same term from both sides keeps the solutions the same, because you could add 2x back to undo it.
$5x - 3 - 2x = 2x + 12 - 2x \quad\text{so}\quad 3x - 3 = 12$
Add 3 to both sides to undo the subtraction of 3. Adding the same number to both sides also keeps the solutions the same.
$3x - 3 + 3 = 12 + 3 \quad\text{so}\quad 3x = 15$
Divide both sides by 3. This step is valid because you are dividing by 3, which is not zero.
$\dfrac{3x}{3} = \dfrac{15}{3} \quad\text{so}\quad x = 5$
Read the final equation. It is true for the number 5 and for no other number. Every step was valid, so the original equation has the same solutions: exactly one.
Check by substituting 5 for x in each side of the original equation. Both sides equal 22, so 5 is a solution.
$\text{left: } 5(5) - 3 = 25 - 3 = 22 \qquad \text{right: } 2(5) + 12 = 10 + 12 = 22$

Answer

The equation $5x - 3 = 2x + 12$ has exactly one solution, $x = 5$ . It is equivalent to $x = 5$ , which is true for only one number, and $x = 5$ makes both sides of the original equation equal $22$ .

Worked example

No solution

How many solutions does $4(x + 2) - x = 3x + 5$ have? Here $x$ may be any real number.

Distribute the 4: 4 times x is 4x and 4 times 2 is 8. The expression you get is equivalent to the one it replaces, which means the two have the same value for every permitted value of x. Replacing one with the other keeps the solutions the same.
$4(x + 2) = 4x + 8 \quad\text{so}\quad 4x + 8 - x = 3x + 5$
Combine the like terms on the left side: 4x minus x is 3x. The new left side is again an equivalent expression.
$3x + 8 = 3x + 5$
Subtract 3x from both sides. The same operation on both sides keeps the solutions the same.
$3x + 8 - 3x = 3x + 5 - 3x \quad\text{so}\quad 8 = 5$
Read the final equation. It has no x in it, and it is false, because 8 does not equal 5. No choice of x can make it true. Every step was valid, so no value of x makes the original equation true either.
Test a value in the original equation, such as 1. The left side is 11 and the right side is 8, which are not equal, as expected. One test cannot prove that there is no solution. The false final equation does that for every value at once.
$\text{left: } 4(1 + 2) - 1 = 12 - 1 = 11 \qquad \text{right: } 3(1) + 5 = 3 + 5 = 8$

Answer

The equation $4(x + 2) - x = 3x + 5$ has no solution. It is equivalent to $8 = 5$ , a false statement that no value of $x$ can make true, so the equation is a contradiction.

Worked example

Infinitely many solutions

How many solutions does $5x - 2(x - 4) = 3x + 8$ have? Here $x$ may be any real number.

Distribute the negative 2 and watch the signs: negative 2 times x is negative 2x, and negative 2 times negative 4 is positive 8. The new left side is an equivalent expression, so the solutions stay the same.
$-2(x - 4) = -2x + 8 \quad\text{so}\quad 5x - 2x + 8 = 3x + 8$
Combine like terms on the left side: 5x minus 2x is 3x.
$3x + 8 = 3x + 8$
Subtract 3x from both sides, the same operation on both sides.
$3x + 8 - 3x = 3x + 8 - 3x \quad\text{so}\quad 8 = 8$
Read the final equation. It has no x in it, and it is true, because 8 equals 8. No choice of x can make it false. Every step was valid, so every permitted value of x makes the original equation true. Here that means every real number.
Test a value in the original equation to catch any slip, such as 3. Both sides equal 17. A test like this supports the work, but the true final equation is what covers every value.
$\text{left: } 5(3) - 2(3 - 4) = 15 - 2(-1) = 15 + 2 = 17 \qquad \text{right: } 3(3) + 8 = 9 + 8 = 17$

Answer

The equation $5x - 2(x - 4) = 3x + 8$ has infinitely many solutions: every real number is a solution. It is equivalent to $8 = 8$ , a true statement whatever $x$ is, so the equation is an identity.

Worked example

When the solution is zero

How many solutions does $7x - 3(x + 2) = 2(x - 3)$ have? Here $x$ may be any real number.

Distribute on each side and watch the signs. On the left, negative 3 times x is negative 3x and negative 3 times 2 is negative 6. On the right, 2 times x is 2x and 2 times negative 3 is negative 6. Each new expression is equivalent to the one it replaces.
$7x - 3x - 6 = 2x - 6$
Combine like terms on the left side: 7x minus 3x is 4x.
$4x - 6 = 2x - 6$
Subtract 2x from both sides, the same operation on both sides.
$4x - 6 - 2x = 2x - 6 - 2x \quad\text{so}\quad 2x - 6 = -6$
Add 6 to both sides, the same operation on both sides.
$2x - 6 + 6 = -6 + 6 \quad\text{so}\quad 2x = 0$
Divide both sides by 2. You are dividing by 2, which is not zero, so the step is valid. Dividing zero by 2 is fine and gives 0. It is dividing by zero that is never allowed.
$\dfrac{2x}{2} = \dfrac{0}{2} \quad\text{so}\quad x = 0$
Read the final equation and check. The final equation still has x in it and is true for exactly one number, zero. Substituting 0 for x in the original equation gives negative 6 on both sides.
$\text{left: } 7(0) - 3(0 + 2) = 0 - 6 = -6 \qquad \text{right: } 2(0 - 3) = 2(-3) = -6$

Answer

The equation $7x - 3(x + 2) = 2(x - 3)$ has exactly one solution, $x = 0$ . It is equivalent to $x = 0$ , which is true for one number only. Zero is a number, so this is one solution, not no solution.

The takeaway. To count the solutions of a linear equation, transform it with valid steps and then read the final equation. Valid steps replace a side with an equivalent expression or do the same operation to both sides, never dividing by zero, so the solutions stay the same.

$x = a$ means exactly one solution. Check it in the original equation.
A false statement such as $8 = 5$ means a contradiction: no solution.
A true statement such as $8 = 8$ means an identity: every permitted value is a solution, so there are infinitely many when $x$ may be any real number.

Next, you will create your own equations with no solutions or infinitely many solutions, and you will find and correct errors in deciding the number of solutions.