Linear Equations in Two Variables

Convert standard form into a readable line

For $3x+2y=12$, subtract $3x$ and divide by 2: $y=-\frac{3}{2}x+6$. The slope is $-\frac{3}{2}$ and the vertical intercept is 6. Setting $y=0$ in the original equation gives $x=4$, so the horizontal intercept is $(4,0)$.

The coefficient of $x$ in standard form is not automatically the slope. If the equation is $Ax+By=C$ with $B\ne 0$, the slope is $-\frac{A}{B}$.

Write a line through a point

A line with slope 2 through $(3,7)$ can be written as $y-7=2(x-3)$. Expanding gives $y=2x+1$. Point-slope form lets you use the given point immediately without guessing the intercept.

If two points are given, first calculate $m=\frac{y_2-y_1}{x_2-x_1}$. A zero denominator means a vertical line, so use $x=c$ rather than forcing slope-intercept form.

Compare parallel and perpendicular lines

Distinct parallel nonvertical lines have the same slope. A line perpendicular to a line of slope $-\frac{3}{2}$ has slope $\frac{2}{3}$, because the slopes multiply to $-1$.

Vertical and horizontal lines are the exception to the reciprocal calculation: $x=4$ is perpendicular to $y=6$. A vertical slope is undefined, not zero.

Test a coordinate in the original equation

The point $(2,3)$ lies on $3x+2y=12$ because $3(2)+2(3)=12$. The point $(3,2)$ does not: it produces 13. Keep the order of coordinates fixed when substituting.

If a problem describes a cost constraint with two items, name each variable and its unit before writing the equation. An equation can be algebraically correct while assigning the prices to the wrong items.

Check your understanding

Sample linear equations in two variables questions

Practice this skill

1. Convert a line, then find both intercepts from the original equation.
2. Explain parallel and perpendicular relationships using slopes.
3. Check every selected ordered pair by substitution.

Print the linear equations in two variables worksheet and worked answers.

Then use a mixed practice module to check whether you can choose this method among other question types.