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C2.3 · Solve and verify multi-term equations with whole numbers and decimals

Learn to solve and verify multi-term equations with whole numbers and decimals through clear examples and targeted practice.

Ontario Grade 7 Mathematics

Algebra

Ontario Grade 7 Mathematics — C2.3

An equation is a math statement that says two amounts are equal. A variable is a letter that stands for a number we do not know yet. To solve an equation is to find the value of the variable that makes the statement true. In this lesson, you will solve equations with more than one term, including equations with decimal numbers. You will also check each answer in the equation you started with.

What you will learn

  • Describe an equation as a statement that two amounts are equal.
  • Solve multi-term equations with whole numbers and decimals by keeping both sides balanced.
  • Verify a solution by substituting it into the original equation.

Think of an equation as a balance

Imagine that the two sides of an equation are two pans on a balanced scale. The amounts in the pans are equal. If you take 4 counters from one pan, you must take 4 from the other pan to keep the scale balanced. In an equation, this means doing the same operation to both sides.
A term is a part of an expression separated by addition or subtraction. In 3x+5=203x+5=20, the left side has two terms: 3x3x and 55. The coefficient is the number multiplied by a variable. In 3x3x, the coefficient is 3. The expression 3x3x means 3 multiplied by the unknown number xx.
It helps to remember opposite operations. Addition and subtraction undo each other. Multiplication and division undo each other. To solve an equation, undo the operations around the variable, one at a time. For an equation such as 3x+5=203x+5=20, first undo adding 5, then undo multiplying by 3.
A multi-term equation has several terms. This lesson focuses on equations where the variable term and number terms are on one side, and the other side is a number. For example, 3x+5=203x+5=20 has a variable term and a number term on the left.
a=b⟹a+c=b+ca=b\Longrightarrow a+c=b+c
  • The equals sign says the two sides have the same value.
  • Do the same operation to both sides to keep the equation balanced.
  • Undo addition or subtraction before undoing multiplication or division.

Solve by undoing operations

Start with a concrete example: three equal groups of an unknown number, plus 5, have a total of 20. To find the unknown, remove the extra 5 from the total. That leaves 15 for the three equal groups. Share 15 equally among the three groups. Each group has 5.
The symbols show the same reasoning. Subtract 5 from both sides of 3x+5=203x+5=20. The equation becomes 3x=153x=15. Then divide both sides by 3. The result is x=5x=5. Each step keeps the two sides equal because the same operation is applied to both.
For a subtraction term, use addition to undo it. In 4x−6=184x-6=18, add 6 to both sides to leave the variable term by itself. Then divide both sides by 4. The goal is to get the variable by itself, not to move a term by changing its sign without explanation.
Decimal numbers follow the same balance steps. In 1.5x+2.4=9.91.5x+2.4=9.9, subtract 2.4 from both sides, then divide by 1.5. When adding or subtracting decimals, line up the decimal points. This keeps tenths with tenths and units with units.
ax+b=c⟹x=c−baax+b=c\Longrightarrow x=\frac{c-b}{a}
  • Undo the number added to or subtracted from the variable term first.
  • Then undo the multiplication by dividing by the coefficient.
  • Line up decimal points when adding or subtracting decimals.

Verify the answer

To verify an answer means to check that it makes the original equation true. Substitute means replace the variable with a number. Put your answer in place of the variable in the equation you started with.
Calculate the left side and the right side separately. If they have the same value, the answer is verified. For example, if x=5x=5 is proposed for 3x+5=203x+5=20, the left side becomes 3(5)+5=203(5)+5=20. The right side is 20, so both sides match.
Check the original equation, not just the last line of your working. This can reveal a mistake made earlier. If the two sides do not match, review your calculations and try the check again.
Try this practice equation: 2.5y+1.5=92.5y+1.5=9. First subtract 1.5 from both sides. Then divide both sides by 2.5. After you find yy, substitute it into the original equation and compare the two sides.
left side=right side\text{left side}=\text{right side}
  • Substitute your answer into the equation you started with.
  • Calculate each side and compare the results.
  • An answer is verified when the two sides are equal.

Worked example

Whole-number terms

Solve and verify 6x−7=296x-7=29.
  1. Undo subtraction
    The 7 is subtracted from the variable term. Add 7 to both sides to undo that subtraction and keep the equation balanced.
    6x−7+7=29+76x-7+7=29+7
  2. Undo multiplication
    The equation now says that 6 times xx is 36. Divide both sides by 6 to find the value of xx.
    6x6=366\frac{6x}{6}=\frac{36}{6}
  3. Verify
    Substitute 6 into the original equation. The left side is 29, which matches the right side.
    6(6)−7=296(6)-7=29
Answer: x=6x=6
Check: The left side is 36−7=2936-7=29. The right side is 29, so the solution is verified.

Worked example

A decimal coefficient and constant

Solve and verify 1.5x+2.4=9.91.5x+2.4=9.9.
  1. Subtract the added amount
    Subtract 2.4 from both sides to remove the number added to the variable term. Line up the decimal points to calculate: 9.9−2.4=7.59.9-2.4=7.5.
    1.5x+2.4−2.4=9.9−2.41.5x+2.4-2.4=9.9-2.4
  2. Divide by the coefficient
    Now 1.51.5 multiplied by xx equals 7.5. Divide both sides by 1.5 to find xx.
    x=7.51.5x=\frac{7.5}{1.5}
  3. Verify
    Substitute 5 into the original equation. The left side equals 9.9, the same as the right side.
    1.5(5)+2.4=9.91.5(5)+2.4=9.9
Answer: x=5x=5
Check: The left side is 7.5+2.4=9.97.5+2.4=9.9. The right side is 9.9, so the solution is verified.

Worked example

A decimal subtraction term

Solve and verify 0.8y−1.6=4.80.8y-1.6=4.8.
  1. Undo subtraction
    Add 1.6 to both sides to undo subtracting 1.6. The equation becomes 0.8y=6.40.8y=6.4.
    0.8y−1.6+1.6=4.8+1.60.8y-1.6+1.6=4.8+1.6
  2. Divide by the coefficient
    Divide both sides by 0.8. Since 6.46.4 divided by 0.80.8 is 8, the value of yy is 8.
    y=6.40.8y=\frac{6.4}{0.8}
  3. Verify
    Substitute 8 into the original equation. The left side is 4.8, which matches the right side.
    0.8(8)−1.6=4.80.8(8)-1.6=4.8
Answer: y=8y=8
Check: The left side is 6.4−1.6=4.86.4-1.6=4.8. The right side is 4.8, so the solution is verified.

Common mistakes and how to avoid them

Changing only one side of an equation.
Correction: Do the same operation to both sides so they remain equal.
Undoing multiplication before removing the added or subtracted number.
Correction: First undo addition or subtraction. Then divide by the coefficient.
Treating a decimal coefficient as a separate number added to the variable.
Correction: A coefficient is multiplied by the variable. For example, 1.5x1.5x means 1.5 multiplied by xx.
Misaligning decimal places when adding or subtracting.
Correction: Line up the decimal points so matching place values are together.
Checking only the last line of the solution.
Correction: Substitute the answer into the original equation and compare both sides.

Lesson summary

  • An equation says that its left and right sides have the same value.
  • Keep an equation balanced by doing the same operation to both sides.
  • Undo addition or subtraction first, then undo multiplication or division.
  • Use the same steps for whole numbers and decimals.
  • Verify a solution by substituting it into the original equation.

Check your understanding

Question 1

Solve 4x+6=264x+6=26. Which value of xx makes the equation true?
  1. 4
  2. 5
  3. 6
  4. 8
Show answer and explanation
5
Subtract 6 from both sides to get 4x=204x=20. Divide by 4 to get x=5x=5. Substitution gives 26 on each side.

Question 2

Solve 2.5y+1.5=92.5y+1.5=9. What is yy?
  1. 2
  2. 2.5
  3. 3
  4. 3.5
Show answer and explanation
3
Subtract 1.5 from both sides to get 2.5y=7.52.5y=7.5. Divide by 2.5 to get y=3y=3. Substitution gives 9 on each side.

Key terms

Equation
A math statement that says two expressions have the same value.
Variable
A letter that stands for a number that is not yet known.
Term
A part of an expression separated from other parts by addition or subtraction.
Coefficient
The number multiplied by a variable.
Substitute
Replace a variable with a number.
Verify
Check an answer by substituting it into the original equation and comparing both sides.

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Published by DoAssignment. This reviewed lesson follows Ontario Grade 7 Mathematics, expectation C2.3. It is a study resource, not an official curriculum publication.

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