Dream School Academy

Linear Equations in One Variable

1) Know the Forms — and When to Use Each

Slope–Intercept Form

y = mx + b shows slope m and intercept b instantly.

Use when: given a graph, or you know the slope and intercept.
Fast to compare slopes/intercepts to classify number of solutions.
Great for quick predictions/interpolations from a line.

Example: If two lines have the same $m$ but different $b$, they’re parallel $\Rightarrow$ no solution.

Point–Slope Form

y - y_1 = m(x - x_1) builds a line from one point and a slope with zero extra work.

Use when: given a slope and one point.
Skip finding the intercept unless you need it; convert to slope–intercept only if necessary.

Example: Through $(4, -1)$ with slope $3$: $y + 1 = 3(x-4)$. Expand only if needed.

Standard Form

Ax + By = C keeps integers tidy and is great for combining equations cleanly.

Use when: you want elimination-friendly integers (avoid fractions).
Handy for quick coefficient comparisons and structured manipulation.

Tip: Clear fractions early (multiply both sides by LCM) to convert to clean integer coefficients.

2) Translate Words → Equation (Blueprint)

Assign variables: Short, meaningful names — $p$ (pay rate), $h$ (hours), $x$ (unknown).
Keyword → operation:
- “plus”, “increased by”, “more than” → +
- “minus”, “decreased by”, “less than” → −
- “times”, “of” (as in % of) → ×
- “per”, “each” → division or multiply by a unit rate
- “is”, “are”, “equals” → =
Units & rates: Match both sides (dollars with dollars, miles with miles). Convert correctly: $(\$/\text{gal}) \div (\text{mi}/\text{gal}) = \$/\text{mi}$.
Build step-by-step: Write math for each sentence chunk, then simplify. Confirm the target before solving — $x$? $x-5$? time? base rate?

Example (gas spend cut): If gas is \$4/gal and the car gets 25 mpg, cost/mi $=\frac{4}{25}$. To reduce weekly spend by \$5: $\frac{4}{25}m = 5 \Rightarrow m = 31.25$ fewer miles.

3) Execute Algebra Cleanly (Advanced Tips)

Clear fractions early: multiply through by LCM to simplify.
Group first, isolate second: combine like terms before moving across the equals sign.
Target expressions directly: if they want $x-5$, isolate $x-5$ (don’t solve for $x$ unless needed).
Control negatives: rewrite subtraction as “add a negative” to reduce sign errors.
Mirror operations: whatever you do to one side, do immediately to the other.
Scan for structure: factor common terms once to save multiple steps later.
Quick system check: if two linear equations appear, compare slopes/intercepts first.

Example (expression target): $2(x-5)+3(x-5)=10\Rightarrow 5(x-5)=10\Rightarrow x-5=2$ (no need to find $x$).

4) Number of Solutions (Fast Classification)

Use slope–intercept $y=mx+b$: different slopes → one solution; same slope and different intercepts → no solution; same slope and same intercepts (or equivalent scaled equation) → infinitely many.

Case	Condition (in $y = mx + b$)	Geometry	# of Solutions	Example Pair
Different slopes	$m_1 \ne m_2$	Lines meet once	One	$y = 2x + 3$ and $y = -x + 4$
Same slope, different intercept	$m_1 = m_2$ and $b_1 \ne b_2$	Parallel, never meet	None	$y = 2x + 3$ and $y = 2x - 5$
Same line	$m_1 = m_2$ and $b_1 = b_2$ (or equivalent scaled equation)	Overlap completely	Infinitely many	$y = 2x + 3$ and $2y = 4x + 6$

Decision Tree

Mini Geometry Sketches

One None Infinitely many

Fast memory: Different slopes → one solution. Same slope + different intercepts → none. Same slope + same intercepts (or scaled equation) → infinitely many.

Case	Condition (in \(y = mx + b\))	Geometry	# of Solutions	Example Pair
Different slopes	\(m_1 \ne m_2\)	Lines meet once	One	\(y = 2x + 3\) and \(y = -x + 4\)
Same slope, different intercept	\(m_1 = m_2\) and \(b_1 \ne b_2\)	Parallel, never meet	None	\(y = 2x + 3\) and \(y = 2x - 5\)
Same line	\(m_1 = m_2\) and \(b_1 = b_2\) (or equivalent scaled equation)	Overlap completely	Infinitely many	\(y = 2x + 3\) and \(2y = 4x + 6\)