Solve addition and multiplication equations with nonnegative rational values

Learning goal: Understand a solution as a value making an equation true, use substitution to test candidates and solve equations of the forms x + p = q and px = q with nonnegative (zero or positive) rational quantities. Explain why the operation used on both sides preserves equality.

The equation x + 2.75 = 8 states that an unknown amount plus 2.75 equals eight. If x represents meters of ribbon already available, 2.75 meters is added and the total becomes eight meters. The unknown names one amount to be found, not permission to choose any value without checking the total.

Test values from the set {4, 5, 5.25, 6}. Substituting four gives 4 + 2.75 = 6.75, which is not eight. Five gives 7.75; 5.25 gives eight; six gives 8.75. Only 5.25 from this set makes the equation true. Testing candidates clarifies the meaning of solution before a general solving procedure is used. The same substitution test decides whether a value satisfies an inequality: 4 satisfies x > 2.5 because 4 > 2.5 is true, and the companion inequality lesson develops that case.

To find the value directly, subtract 2.75 from both sides: x + 2.75 − 2.75 = 8 − 2.75. The added amount is undone, leaving x = 5.25. Subtracting equal amounts from equal quantities preserves equality. Subtracting from only one side would generally destroy the relationship.

The tape diagram shows the eight-meter total split into an unknown segment and a 2.75-meter segment. Removing the known segment leaves the unknown 5.25-meter length. The model's lengths use the actual 5.25:2.75 ratio. The complete tape remains the original eight-meter total even while one segment is highlighted.

A multiplication equation can describe equal groups. Three equal bags have total mass 7.5 kilograms. Let x be kilograms per bag; then 3x = 7.5. Divide both sides by three to obtain x = 2.5 kilograms per bag. The equation's three groups and its total determine the operation. Check 3 × 2.5 = 7.5.

The multiplier can be fractional. If (3/4)x = 6, divide both sides by 3/4, giving x = 6 ÷ (3/4) = 8. Multiplication confirms (3/4) × 8 = 6. A tape divided into four equal parts would show three parts totaling six, so each part is two and the entire four-part amount is eight.

An addition equation can also involve fractions: x + 5/6 = 2 gives x = 2 − 5/6 = 12/6 − 5/6 = 7/6. The solution is one and one sixth, a nonnegative rational number. It need not be a whole number merely because the equation looks short.

Solving and checking are different steps. An operation can suggest a candidate solution, but substitution into the original equation verifies it. If a learner obtains x = 10.75 for x + 2.75 = 8 by adding instead of subtracting, substitution gives 13.5 rather than eight and exposes the error.

Some equations have no solution in a stated set. x + 5 = 3 has no nonnegative solution because adding five to a nonnegative number cannot produce three. This lesson does not force a negative answer into its set of nonnegative values. The stated set of allowed values matters when deciding which values are permissible.

For px = q with p = 0, division by p is unavailable. If q is also zero, every allowed x satisfies 0x = 0. If q is positive, no x satisfies 0x = q. The ordinary divide-both-sides procedure assumes a nonzero multiplier. Recognizing that assumption avoids treating division by zero as a valid step.

Worked contextual equation: A fictional material costs $1.25 per meter and the total is $8.75. With x meters, 1.25x = 8.75, so x = 7 meters. Check the original cost relationship, 1.25 × 7 = 8.75. Prices are invented for arithmetic practice.

Worked candidate test: For x + 5/6 = 2, the candidate x = 1 fails because the left side is 1 5/6. The candidate x = 7/6 succeeds because 7/6 + 5/6 = 12/6 = 2.

Guided modeling: Use a tape for an additive equation and equal groups for a multiplicative equation. State what the variable measures, perform the same valid operation on both sides and check the original statement.

Independent performance: Solve addition and multiplication equations with whole-number, decimal and fractional values. Include a candidate set with no valid member and a zero-multiplier case. Write one equation from a real-world scenario, explain the solution's unit and its set of allowed values, and verify by substitution. Do not replace the equality explanation with an unexplained instruction to move a number across the equals sign.