Linear equations in two variables

Linear equations in two variables express relationships between two unknown quantities, usually $x$ and $y$, where each variable is raised to the first power. These relationships can be represented algebraically or visually as straight lines on a coordinate plane.

Understanding Slope-Intercept Form: $y = mx + b$

The slope is equal to $m$.

• Slope describes the rate of change between the two variables $x$ and $y$.

• On a graph, slope represents the rise over run ($\frac{\Delta y}{\Delta x}$), determining the direction and steepness of the line.

• In a word problem, slope represents the unit rate of change of $y$ with respect to $x$.

• Words like "per", "each", and "rate" are clues that point to the slope.

• Parallel lines have equal slopes ($m_1 = m_2$), while perpendicular lines have negative reciprocal slopes ($m_1 \cdot m_2 = -1$).

The $y$-intercept is equal to $b$.

• The $y$-intercept is the point $(0, b)$ where the line crosses the $y$-axis.

• On a graph, it shows the value of $y$ when $x = 0$.

• In a word problem, the $y$-intercept represents the initial value, baseline cost, or starting amount before any changes in $x$ occur.

• Words like "initial", "base fee", and "deposit" are clues that point to the $y$-intercept.

Understanding Standard Form: $Ax + By = C$

Standard form is written as $Ax + By = C$, where $A$, $B$, and $C$ are usually integers (with $A \ge 0$).

• Standard form is best used for scenarios where two different items or rates combine to equal a fixed total.

• In a word problem, $x$ and $y$ represent quantities of two different categories, while $A$ and $B$ represent their unit rates or costs.

• The slope of a line in standard form can be calculated directly as $m = -\frac{A}{B}$.

• To find the $x$-intercept, set $y = 0$ to get $( \frac{C}{A}, 0 )$.

• To find the $y$-intercept, set $x = 0$ to get $( 0, \frac{C}{B} )$.

Point-Slope Form: $y - y_1 = m(x - x_1)$

Point-slope form uses a known point $(x_1, y_1)$ and the slope $m$.

• Point-slope form is useful when given the slope and any single point on the line, or when given two points from which the slope is first calculated.

• In word problems, $(x_1, y_1)$ represents a specific known state or data point rather than the starting $y$-intercept.

• Rearranging $y - y_1 = m(x - x_1)$ by expanding and isolating $y$ converts the equation directly into slope-intercept form $y = mx + b$.

1. Calculate slope quickly using the slope formula or standard form shortcut. Example: What is the slope of the line passing through $(2, 5)$ and $(6, 13)$?

   • Slope formula: $m = \frac{13 - 5}{6 - 2} = \frac{8}{4} = 2$.

   • Shortcut for standard form $3x + 4y = 12$: $m = -\frac{A}{B} = -\frac{3}{4}$.

2. Use parallel and perpendicular slope rules to find line equations. Example: What is the slope of a line perpendicular to $y = -\frac{2}{3}x + 4$?

   • Parallel slope: Same slope, so $m = -\frac{2}{3}$.

   • Perpendicular slope: Flip the fraction and change the sign (negative reciprocal), so $m = \frac{3}{2}$.

3. Find intercepts instantly by setting the opposite variable to zero. Example: Find the $x$-intercept of $5x - 2y = 20$.

   • Set $y = 0$: $5x - 2(0) = 20 \implies 5x = 20 \implies x = 4$.

   • $x$-intercept point: $(4, 0)$.

4. Translate two-variable word problems into standard form. Example: Tickets cost $\$8$ for adults ($x$) and $\$5$ for children ($y$). Total sales were $\$400$.

   • Adult sales term: $8x$

   • Child sales term: $5y$

   • Equation: $8x + 5y = 400$

5. Match table data to linear equations using two sample points. Example: A table gives points $(1, 7)$ and $(3, 13)$. Which equation fits the data?

   • Calculate slope: $m = \frac{13 - 7}{3 - 1} = \frac{6}{2} = 3$.

   • Find $y$-intercept using $(1, 7)$: $7 = 3(1) + b \implies b = 4$.

   • Equation: $y = 3x + 4$.

6. Interpret geometric properties of horizontal and vertical lines. Example: What is the equation and slope of a vertical line passing through $(5, -3)$?

   • Vertical line equation: $x = 5$ (slope is undefined).

   • Horizontal line equation: $y = -3$ (slope is $0$).