Algebra / Linear Equations Two Variables
Linear Equations in Two Variables
A linear equation in two variables can be written in forms such as $ax+by=c$ or $y=mx+b$. A solution is an ordered pair $(x,y)$ that makes the equation true.
Least you need to know
- An ordered pair is a solution if substituting its $x$- and $y$-values makes the equation true.
- The $x$-intercept occurs when $y=0$; the $y$-intercept occurs when $x=0$.
- A single linear equation in two variables usually has infinitely many real-number solutions.
Key notation
- $(x,y)$ — An ordered pair whose first coordinate is $x$ and second coordinate is $y$
- $ax+by=c$ — A common standard form for a linear equation in two variables
Worked example
- Consider $3x+2y=18$.
- If $x=4$, substitute 4 for $x$: $3(4)+2y=18$.
- Simplify: $12+2y=18$.
- Subtract 12: $2y=6$.
- Divide by 2: $y=3$.
Common mistakes
- In an ordered pair $(x,y)$, the first coordinate is $x$ and the second is $y$.
- For the $x$-intercept, set $y=0$, not $x=0$.
- When rewriting in slope-intercept form, isolate $y$ completely.
How to recognize it
- The equation contains two variables, usually $x$ and $y$.
- The problem may ask whether a point lies on a line or ask for an intercept.
- The problem may give one coordinate and ask for the other.
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