Solve and check linear equations

Which statement describes the equation shown?

An equation is a statement of equal values

An expression names a quantity. An equation uses an equals sign to state that two quantities have the same value

A solution is a value of the unknown that makes the equation true

Equality works in either direction

The equations in this topic have positive whole-number solutions. We will find the value and then check it

Equation

\(\displaystyle x+5=12\)

Proposed value

\(\displaystyle x=7\)

For the equation shown, why is the proposed value a solution?

Apply the same reversible operation to both sides

If two quantities are equal, adding the same amount to both or subtracting the same amount from both keeps them equal

Multiplying or dividing both sides by the same nonzero number also preserves the equality

We subtract five from both sides to undo the added five. We do not simply delete a term from one side

Never divide by zero

Both rows start from twelve equals twelve. The top row removes s from both sides. The bottom row removes s from the left side only. Change s to compare the results

When five is removed, which row still has equal values?

Undo an added constant

Find the value of \(x\) that makes this equation true

Subtract 6 from both sides

Check by substituting \(x=9\) into the original equation

Both sides have the same value, so the solution is correct

Subtract the same amount from both sides

Undo subtraction with addition

If a constant has been subtracted from the unknown, add that constant to both sides

Check the original subtraction, not the rearranged final line

Restore the amount that was subtracted

Find the value of \(t\) that makes this equation true

Add 5 to both sides

Check by substituting \(t=8\) into the original equation

Both sides have the same value, so the solution is correct

Use the inverse of subtraction

Undo equal groups with division

A coefficient multiplies the unknown. Divide both sides by that nonzero coefficient to find one copy

Subtracting four would not undo multiplication by four

Find the value of one equal group

Find the value of \(x\) that makes this equation true

Divide both sides by 5

Check by substituting \(x=7\) into the original equation

Both sides have the same value, so the solution is correct

Divide by the coefficient

Undo division with multiplication

If the unknown has been divided by a nonzero constant, multiply both sides by that constant

The original division is the check

Recover the total from one share

Find the value of \(t\) that makes this equation true

Multiply both sides by 4

Check by substituting \(t=8\) into the original equation

Both sides have the same value, so the solution is correct

Multiply both sides by the divisor

Undo operations in reverse order

In three times x plus five, the forward process multiplies x by three and then adds five

To recover x from the result, undo the added five first and then undo the multiplication by three

Each step applies the inverse operation to both sides. It is not a rule to move a symbol across an equals sign without a reason

Remove the fixed term before dividing

Find the value of \(n\) that makes this equation true

Subtract 3 from both sides

Divide both sides by 4

Check by substituting \(n=7\) into the original equation

Both sides have the same value, so the solution is correct

Undo addition and then multiplication

Undo a subtracted term before dividing

Find the value of \(p\) that makes this equation true

Add 5 to both sides

Divide both sides by 3

Check by substituting \(p=7\) into the original equation

Both sides have the same value, so the solution is correct

Add the missing amount, then divide

Which first step preserves the equality and starts to isolate x?

A bracket can be treated as one multiplied quantity

In three times the whole quantity x plus two, the outer operation is multiplication by three

Divide both sides by three to leave the bracketed quantity

Then subtract two from both sides

Expanding first is another valid route, provided the multiplier applies to every term

Both valid routes give the same solution

Undo the outside multiplication first

Find the value of \(x\) that makes this equation true

Divide both sides by 4

Subtract 3 from both sides

Check by substituting \(x=7\) into the original equation

Both sides have the same value, so the solution is correct

Keep the bracket together while dividing

A proposed solution must make the original equation true

A claim is not enough. Replace the unknown in the original equation with the proposed value and calculate each side separately

Five is not a solution because the sides differ

Checking only the final rearranged line can repeat an earlier mistake. Use the original equation

In the verification tasks, enter both side values and then decide whether they are equal

Calculate both sides for a claimed value

Proposed value: \(x=6\)

Calculate each side of the original equation using the proposed value

Substitute the proposed value into the original equation, not into a rearranged equation

The sides are equal, so the proposed value is a solution

Does the proposed value make the original equation true?

Test a candidate that may satisfy the equation

Calculate before deciding whether the claim works

Substitute inside the original bracket

Interpret the unknown before solving

A fictional activity charges four dollars per person and one fixed three-dollar fee. The total is thirty-one dollars. Let n count the people

Remove the fixed fee, then divide by the per-person charge

The result means seven people, not seven dollars

The check recovers the entire stated cost

Find a count from a total and a fixed charge

Recover a whole length from equal pieces

Use a supplied perimeter equation to find a length

Original equation

\(\displaystyle 3n+5=32\)

Proposed value

\(\displaystyle n=9\)

In a ticket problem, n counts tickets. A learner obtains the proposed value. Which is the complete interpretation and check?

Solve by preserving equality, then verify

An equation requires the entire left side and right side to have equal values

Apply the same reversible operation to both sides. Use addition to undo subtraction and division to undo multiplication

For two-step equations, undo the operations in reverse order. A scaled bracket can be treated as one quantity

Check a proposed value by substituting it into both sides of the original equation and comparing the results

State what the solution means in the given situation and use its stated unit