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MathChapter 2: Solving Linear Equations

Notes

Solving for a Specific Variable Notes

Rearrange formulas and literal equations by treating non-target symbols as constants, factoring the target when needed, and preserving whole expressions.

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A literal equation contains two or more variables and describes a general relationship. Solving for a specific variable means rearranging the equation so that the target appears alone on one side. You are not expected to calculate a number unless values are given; all non-target symbols are treated as constants.

Learning objectives

  • Identify the target variable and treat every other symbol as a constant.
  • Use inverse operations to isolate a target that appears once.
  • Clear fractions and preserve grouped expressions when dividing.
  • Factor the target variable when it appears in multiple terms or on both sides.
  • Rearrange common area, temperature, rate, linear, and geometry formulas.

Literal equations and the target mindset

Literal equation
An equation containing two or more variables, usually expressing a formula or general relationship. For example, \(d=rt\) relates distance, rate, and time.

Target → isolate target term → factor if needed → divide → audit

  1. Name the target

    Circle the variable requested after words such as ‘solve for.’

  2. Simplify structure

    Clear convenient fractions, distribute only when useful, and combine like terms.

  3. Isolate target-containing terms

    Move every term containing the target to one side and all other terms to the other.

  4. Factor the target

    If the target appears in more than one term, factor it out before dividing.

  5. Divide the complete side

    Divide by the target's full coefficient or factored expression, noting nonzero restrictions.

  6. Audit

    Substitute the rearranged expression into the original structure or reverse the operations mentally.

Target variable appearing once

Worked example

Solve a linear form for its input

Solve \(y=mx+b\) for \(x\).

  1. Identify target

    The target is \(x\); treat \(y,m,b\) as constants.

  2. Remove the added constant

    Subtract \(b\) from both sides: \(y-b=mx\).

  3. Remove the coefficient

    Divide the entire left side by \(m\), assuming \(m\ne0\).

  4. Audit grouping

    The numerator must remain \(y-b\); dividing only \(y\) would not undo multiplication of the whole \(x\)-term.

\(x=\frac{y-b}{m}\), for \(m\ne0\).

Rearranging familiar formulas

Original formula-rearrangement examples
RelationshipSolve forRearranged formRestriction when relevant
Distance \(d=rt\)\(t\)\(t=d/r\)\(r\ne0\)
Rectangle area \(A=LW\)\(W\)\(W=A/L\)\(L\ne0\)
Triangle area \(A=bh/2\)\(h\)\(h=2A/b\)\(b\ne0\)
Celsius \(C=\frac59(F-32)\)\(F\)\(F=\frac95C+32\)None beyond real-valued variables
Worked example

Undo a grouped temperature formula

Solve \(C=\frac59(F-32)\) for \(F\).

  1. Undo the fraction

    Multiply both sides by \(\frac95\): \(\frac95C=F-32\).

  2. Undo subtraction

    Add \(32\) to both sides: \(\frac95C+32=F\).

  3. Present target first

    Rewrite with \(F\) on the left.

\(F=\frac95C+32\).

Target in a denominator

When the target is in a denominator, first clear the denominator by multiplying both sides by the target expression, while recording that it cannot be zero. Then isolate the target normally.

Worked example

Solve a rate formula for a denominator variable

Solve \(R=\frac{D}{t}\) for \(t\).

  1. Record restriction

    The original formula requires \(t\ne0\).

  2. Clear the denominator

    Multiply both sides by \(t\): \(Rt=D\).

  3. Isolate target

    Divide by \(R\), assuming \(R\ne0\): \(t=D/R\).

\(t=\frac DR\), with the relevant nonzero conditions.

Target appearing in more than one term

Worked example

Factor the target from two terms

Solve \(P=xr+xs\) for \(x\).

  1. Identify target terms

    Both terms on the right contain \(x\).

  2. Factor

    Use the distributive property in reverse: \(P=x(r+s)\).

  3. Divide

    Divide both sides by \(r+s\), assuming \(r+s\ne0\).

\(x=\frac{P}{r+s}\).
Worked example

Target on both sides

Solve \(ax+b=cx+d\) for \(x\).

  1. Collect target terms

    Subtract \(cx\): \(ax-cx+b=d\).

  2. Collect constants

    Subtract \(b\): \(ax-cx=d-b\).

  3. Factor target

    Write \(x(a-c)=d-b\).

  4. Divide

    If \(a-c\ne0\), divide by \(a-c\).

\(x=\frac{d-b}{a-c}\), provided \(a\ne c\).

Grouped and fractional literal equations

Worked example

Isolate a target inside a factored expression

Solve \(Q=k(3x-r)+s\) for \(x\).

  1. Remove outside constant

    Subtract \(s\): \(Q-s=k(3x-r)\).

  2. Remove outside factor

    Divide by \(k\): \((Q-s)/k=3x-r\), assuming \(k\ne0\).

  3. Remove inner constant

    Add \(r\): \((Q-s)/k+r=3x\).

  4. Isolate target

    Divide the complete left side by \(3\).

\(x=\frac{Q-s+kr}{3k}\), equivalently \(x=\frac13\left(\frac{Q-s}{k}+r\right)\), for \(k\ne0\).

Common mistakes and SAT strategy

Mini check

Check your understanding

Solve \(M=3x+xy\) for \(x\).

  1. \(x=M/(3+y)\)
  2. \(x=M/3+y\)
  3. \(x=(M-y)/3\)
  4. \(x=3M/y\)
Show answer and explanation

Answer: \(x=M/(3+y)\)

Factor \(x\) from the right: \(M=x(3+y)\). Divide by the complete factor \(3+y\), assuming it is nonzero.

Key takeaways

Key takeaways

What to remember

  • Circle the target variable and treat every other symbol as a constant.
  • Undo operations around a target that appears once from outside inward.
  • Keep multi-term numerators grouped when dividing a whole side.
  • If the target appears in multiple terms, collect those terms and factor the target before dividing.
  • Record nonzero restrictions introduced by denominators or final division.
  • A symbolic rearrangement is a complete answer when no numerical values are given.
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Put these notes into practice

Apply the ideas with SAT-style questions, then reinforce key details with flashcards.