What does it mean for two sides of an equation to be equal?
Quick answer
The equal sign says that the expression on its left and the expression on its right have the same value. An equation is a statement making that claim, not merely a cue to calculate the left side and write an answer. For example, 3 + 4 = 7 is true because both sides are worth 7; 3 + 4 = 8 is false because they are not.
The question, the details, the next step.
What does it mean when two sides of an equation are equal? I tend to read the equals sign as “now write the answer.”
Think of the equal sign as saying “has the same value as.” An equation connects two expressions and asserts that they are equal; it is not simply an instruction to calculate one side. [S1]
For instance, in 3 + 4 = 7, the left expression, 3 + 4, and the right expression, 7, both have the value 7. The statement is true. In 3 + 4 = 8, the sides have different values, so the statement is false.
A balance scale is a useful picture: if the two sides represent equal amounts, the scale balances. The picture is about equal value, not necessarily identical appearance. The expressions 3 + 4 and 7 look different, but they represent the same amount.
So if the two sides look different, how can I check whether they really match?
Work out the value of each side, keeping each side separate, then compare the results. For example, consider 2 × 5 = 6 + 4. The left side is 10; the right side is also 10. The equation is true even though one side uses multiplication and the other uses addition.
For a simple equation with a letter, x + 3 = 8 makes a claim about the value of x. Try x = 5: the left side becomes 5 + 3, which is 8, matching the right side. Try x = 4: the left becomes 7, which does not match 8. So 5 makes this equation true, while 4 does not.
The balance picture helps here too: changing a value on one side can make the sides stop matching. To check a proposed value, substitute it for the letter and compare the resulting amounts.
What is the misconception I should watch for when I see an equals sign in a longer expression?
Watch for treating = as a one-way arrow meaning “do the calculation and put the answer next.” An equals sign relates what is on both sides. That is why a statement such as 10 = 6 + 4 can be true: the order of the sides does not change their values.
Also check that there is a complete statement on each side. For example, 2 + 3 = 5 + 1 = 6 may be read as two comparisons in a row: the first says 5 equals 6, which is false. If you intend to show two separate calculations, write them on separate lines instead:
2 + 3 = 5
5 + 1 = 6
The key question is not “What answer comes after the equals sign?” but “Do the expressions on both sides have the same value?”
Can you give me a short routine I can use when checking an equation on my own?
Use this routine:
- Point to the left side and identify the whole expression.
- Point to the right side and identify the whole expression.
- If there is a letter, substitute the value you are testing.
- Work out each side separately.
- Compare the results: if they match, the equation is true for that value; if they differ, it is not.
Worked check for 4 + 2 = 3 × 2: left side, 4 + 2, is 6; right side, 3 × 2, is 6. Both sides match, so the equation is true. This routine checks whether a particular equation or proposed value works; it does not, by itself, explain every method for finding unknown values.
A reusable note can look like this:
Equation: [write the equation] Value tested, if any: [write the value] Left side: [calculate] Right side: [calculate] Do they match? [yes/no] Conclusion: [true / false for this value]
A practical checklist
Read
=as “has the same value as,” not as an instruction to write an answer.Identify the complete expression on each side of the equal sign.
Substitute a proposed value for any letter before checking.
Work out the two sides separately, then compare their values.
If showing separate calculations, put them on separate lines to avoid joining unequal statements.
A few more useful answers
Can the two sides be written in a different order?
Yes. If both expressions have the same value, either can appear on the left or the right. For example, 7 = 3 + 4 is true for the same reason as 3 + 4 = 7.
Does a true equation tell me the value of a letter automatically?
Not necessarily. Checking a proposed value shows whether it makes the equation true. Finding an unknown value may require additional steps.
Can an equation be false?
Yes. An equation makes a claim that the two sides have the same value, and that claim can be true or false. For example, 3 + 4 = 8 is false because the sides have different values.
Sources
References used for this guide.
- [S1] OpenStax Elementary Algebra 2e: Use the Language of Algebra openstax.org · Checked