Solve a linear equation

Predict the expansion when the outside factor is negative

The tiles represent k(x − 3). Filled tiles are added and hollow tiles are subtracted. Change the outside factor from 1 to −1, −2 and −3, then compare 0 and positive factors. Track both the x coefficient and the constant term

What happens to the constant term when k is negative?

Keep whole groups intact

Distribute an outside factor to every term, including signs. A minus sign before a bracket is a factor of −1. When clearing numerical denominators, multiply every term on both sides by a common multiple. Then collect unknown terms and constants, solve and substitute into the original

1. 1Negative bracket Both signs change when a bracket is subtracted \(\displaystyle -(x-4)=-x+4\)
2. 2Fractions on both sides If each side is one fraction, multiplying by both non-zero numerical denominators gives the cross products \(\displaystyle \frac{x+1}{3}=\frac{x-2}{2}\Rightarrow 2(x+1)=3(x-2)\)
3. 3A sum of fractions A sum has several terms. Apply the common multiplier to each term; do not cross multiply just the nearest terms

Equation or model

\(\displaystyle -3(2x-5)=21\)

Solve without losing the negative outside factor

Equation or model

\(\displaystyle 2(x+4)-3(x-1)=18\)

Expand both brackets and collect like terms

Equation or model

\(\displaystyle 4(x-2)=2(3x+1)\)

Solve the equation with brackets on both sides

Which expansion preserves the value of the subtracted bracket?

Subtracting 3x from both sides gives 2 = 7. How many solutions does the original equation have?

Expanding the left side gives 2x + 6 = 2x + 6. What is the solution set?

Equation or model

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

Clear the denominator of the whole numerator and solve exactly

Equation or model

\(\displaystyle \frac{2(x-1)}{3}=5\)

Clear the denominator before undoing the bracket factor

Multiply every term by 6. Which equation is equivalent?

Equation or model

\(\displaystyle \frac{x+1}{3}-\frac{x-2}{2}=1\)

Use the common denominator 6, then solve

Why is cross multiplying only the first fraction and the right side invalid here?

Equation or model

\(\displaystyle \frac{x}{3}+\frac12=\frac{5x}{6}-1\)

Multiply all four terms by 6 and solve exactly

The exact solution is shown below. No rounding is requested. Which answer preserves it?