Solve and verify signed linear equations

Which statement describes a solution of this equation?

Preserve equality rather than moving symbols without a reason

An equation joins two expressions whose values must be equal. Apply the same reversible operation to both complete sides

Addition and subtraction can be reversed. Multiplication and division preserve an equivalent equation when the number used is nonzero

Checking uses the original equation, including every constant, sign, bracket and fraction bar

The left side equals the original right side, so negative four is a solution

Which step preserves the solutions of the given equation?

Solve a signed two-step equation and check both sides

Find the exact value of \(x\) as an integer or reduced fraction

Subtract 5 from both sides

Divide both sides by 2

Check by substituting \(x=-4\) into the original equation

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

Use your solved value in the original equation and calculate each side separately. Enter each result as an integer or a reduced fraction

Substitute the solved value \(x=-4\) into both sides of the original equation

Both original sides have the same value, so the solution is verified

Divide by a negative coefficient without losing the sign

Zero is a possible value of the unknown

Add seven to both sides

Divide both sides by the nonzero coefficient four

The solution is zero, not no solution. Zero is a number that can be substituted and checked

Distinguish a zero solution from no solution

Keep a fractional solution exact

There is no requirement for the solution to be an integer. Keep the exact fraction rather than rounding

Use a reduced fraction when a task requests one. A negative fraction keeps its negative sign

The answer controls in this topic ask for exact integers or fractions rather than rounded decimal estimates

Construct an exact fractional solution

Find the exact value of \(x\) as an integer or reduced fraction

Subtract 1 from both sides

Divide both sides by 2

Check by substituting \(x=\frac{3}{2}\) into the original equation

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

Use your solved value in the original equation and calculate each side separately. Enter each result as an integer or a reduced fraction

Substitute the solved value \(x=\frac{3}{2}\) into both sides of the original equation

Both original sides have the same value, so the solution is verified

Keep a negative fractional solution exact

Undo the outside multiplication or expand every term

The whole bracket is multiplied by negative three. Dividing both sides by negative three gives a shorter route

Expanding first is also valid if both terms are multiplied

The original bracketed equation, not only the expanded line, is used for verification

Solve a negatively scaled bracket

Find the exact value of \(x\) as an integer or reduced fraction

Divide both sides by -3

Subtract 4 from both sides

Check by substituting \(x=-6\) into the original equation

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

Use your solved value in the original equation and calculate each side separately. Enter each result as an integer or a reduced fraction

Substitute the solved value \(x=-6\) into both sides of the original equation

Both original sides have the same value, so the solution is verified

Allow an exact fractional value inside a bracket

A fraction bar can group a whole numerator

The entire numerator three x minus two is divided by four. Multiply both complete sides by four

Do not multiply just one term in the numerator or just one side of the equality

Clear the denominator of a complete numerator

Find the exact value of \(x\) as an integer or reduced fraction

Multiply both sides by 4

Add 2 to both sides

Divide both sides by 3

Check by substituting \(x=2\) into the original equation

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

Use your solved value in the original equation and calculate each side separately. Enter each result as an integer or a reduced fraction

Substitute the solved value \(x=2\) into both sides of the original equation

Both original sides have the same value, so the solution is verified

Clear a fraction bar while retaining a negative coefficient

A constant outside the fraction has a different scope

Here only three x is divided by two. The subtraction of two happens outside the fraction

Add two to both sides before clearing the denominator

Multiplying the original equation by two first is also valid only if every complete-side term is multiplied

Remove the outside constant before the denominator

Find the exact value of \(x\) as an integer or reduced fraction

Add 2 to both sides

Multiply both sides by 2

Divide both sides by 3

Check by substituting \(x=2\) into the original equation

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

Use your solved value in the original equation and calculate each side separately. Enter each result as an integer or a reduced fraction

Substitute the solved value \(x=2\) into both sides of the original equation

Both original sides have the same value, so the solution is verified

Solve a negative fractional coefficient equation

Original equation

\(\displaystyle \frac{3x}{4}+2=5\)

Which equation results from multiplying both sides by four?

Collect unknown terms using an equal operation on both sides

Subtract two x from both complete sides

Add four to both sides, then divide both sides by three

Both sides now require substitution calculations

Subtracting five x instead is another valid route; it gives a negative remaining coefficient but the same solution

Solve a fractional equality with unknowns on both sides

Find the exact value of \(x\) as an integer or reduced fraction

Subtract 2x from both sides

Add 4 to both sides

Divide both sides by 3

Check by substituting \(x=\frac{11}{3}\) into the original equation

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

Use your solved value in the original equation and calculate each side separately. Enter each result as an integer or a reduced fraction

Substitute the solved value \(x=\frac{11}{3}\) into both sides of the original equation

Both original sides have the same value, so the solution is verified

Collect terms when the resulting coefficient is negative

Find and verify zero with unknowns on both sides

Expand both brackets before collecting unknowns

Each outside factor multiplies every term in its own bracket

A negative outside factor changes the sign of each term when expanding. Keep that multiplication separate from moving terms between sides

Expand both sides and check the original brackets

Find the exact value of \(x\) as an integer or reduced fraction

Multiply both terms in the left bracket by 3

Multiply both terms in the right bracket by 2

Subtract 2x from both sides

Subtract 6 from both sides

Divide both sides by 1

Check by substituting \(x=4\) into the original equation

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

Use your solved value in the original equation and calculate each side separately. Enter each result as an integer or a reduced fraction

Substitute the solved value \(x=4\) into both sides of the original equation

Both original sides have the same value, so the solution is verified

Expand a negative factor and retain a fractional result

Check a signed solution in both original brackets

A familiar-looking final line is not a substitute for verification

A learner claims that x equals one solves the following equation

The claim fails because the original sides differ

Solving the original equality gives negative two x plus six equals three x plus three, so five x equals three

Equation

\(\displaystyle -3x+2=5\)

Proposed value

\(\displaystyle x=1\)

Calculate the original left side using the proposed value. The original right side is five, so compare after calculating

Multiply negative three by the candidate one, then add two

Graph the two sides as separate relations

Plot y equal to the left expression and y equal to the right expression using the same axes and scale

At the intersection, the two expressions have the same output. The solution of the original equation is its horizontal x-coordinate

The common y-value seven verifies the equality, but it is not the requested x solution

A graph reading can be approximate. Use algebra or substitution to establish an exact value when the grid does not determine it precisely

Equation

\(\displaystyle 3x-2=x+4\)

Graphs

\(\displaystyle y=3x-2,\quad y=x+4\)

Find x where these two graphs meet, then verify the value in the original equality

Read the horizontal coordinate of P, not its vertical coordinate

Equation

\(\displaystyle 3x-2=x+4\)

Intersection

\(\displaystyle P=(3,7)\)

The graphs of the two sides meet at P. Which statement answers the original equation?

Change the added constant in the displayed equation. The moving point marks where the graphs of its left and right sides meet

Which statement describes the moving intersection for every permitted b?

Check what remains if the unknown terms cancel

Subtracting two x from both sides leaves a false statement

No real input satisfies this equation. Its two distinct parallel graphs do not intersect

Expanding the left side gives the same expression as the right side

Every real input satisfies this equality, and the two graphs coincide

Do not divide by the remaining coefficient if it is zero. Decide whether the remaining constant statement is true or false

The generated solve-and-check tasks in this topic all have a unique solution; these examples explain why that is not true of every equation

What is the solution set of this equation?

What is the solution set of this equation?

Explain and compare two solution methods

Independent written and digital task, not automatically marked

Create a signed linear equation with a unique fractional solution. Explain each reversible operation used to solve it

Calculate both original sides at your exact solution, then graph the two expressions using a digital graphing tool

Choose a viewing window that contains the intersection. Compare its horizontal coordinate with your algebraic result and explain any rounding in the graphical display

Create a different equation with no solution and one that is true for every real input. Explain the cancellation and graph behaviour in each

Review the equations, graphs and explanations against these instructions or with a teacher. The automatic tasks check the supplied equations and values, not unrestricted creation or written method quality

Use equality, exact values and original-side checks together

Apply the same reversible operation to both complete sides, and never divide by zero

A unique solution can be negative, zero or fractional. Keep exact values unless approximation is requested

Expand every term in a bracket and respect which terms are covered by a fraction bar

When unknowns occur on both sides, collect them by adding or subtracting the same term from both sides

Verify in the original equation by calculating both side values, not only by inspecting a final rearranged line

Graphing the two expressions locates solutions at intersection x-coordinates; the common y-value helps verify equality

If unknown terms cancel, a false constant statement means no solution and a true one means every permitted real input is a solution