Lines and systems

Points and intercepts

An ordered pair \((x, y)\) is on a line if it makes the equation true. x-intercept: set \(y = 0\). y-intercept: set \(x = 0\).

Example: Is \((2, 5)\) on \(y = 3x - 1\)? \(3(2) - 1 = 5\). Yes.

Example: \(2x + 3y = 12\): x-intercept \((6, 0)\), y-intercept \((0, 4)\).

Slope

Slope = rise over run: \(m = \dfrac{y_2 - y_1}{x_2 - x_1}\).

Example: \((-1, 4)\) and \((3, -4)\): \(m = \dfrac{-4 - 4}{3 - (-1)} = \dfrac{-8}{4} = -2\).

Positive: rises left to right. Negative: falls. Zero: horizontal. Undefined: vertical.

Watch out: keep the same point first on top and bottom.

Horizontal and vertical lines

\(y = k\) is horizontal with slope 0. \(x = h\) is vertical with undefined slope.

Example: \(y = -2\) is flat through \((0, -2)\); \(x = 3\) is straight up and down through \((3, 0)\).

Slope-intercept form

Slope-intercept form: \(y = mx + b\), slope \(m\), y-intercept \((0, b)\). To graph: plot \((0, b)\), then move rise and run.

Example: \(4x - 2y = 6\): \(-2y = -4x + 6\), so \(y = 2x - 3\): \(m = 2\), \(b = -3\).

Point-slope form

Point-slope form: \(y - y_1 = m(x - x_1)\) for slope \(m\) through \((x_1, y_1)\).

Example: Slope 3 through \((2, -1)\): \(y + 1 = 3(x - 2)\), so \(y = 3x - 7\).

A line through two points

Find the slope first, then use point-slope form with either point.

Example: \((1, 2)\) and \((3, 8)\): \(m = \dfrac{6}{2} = 3\); \(y - 2 = 3(x - 1)\), so \(y = 3x - 1\).

Standard form

Standard form: \(Ax + By = C\). Its slope is \(-\dfrac{A}{B}\) (when \(B \ne 0\)). Good for finding intercepts.

Example: \(2x + 3y = 12\) has slope \(-\dfrac{2}{3}\).

Parallel and perpendicular

Parallel: same slope. Perpendicular: negative reciprocal slopes, \(m_1 \cdot m_2 = -1\) (flip and change the sign).

Example: Through \((4, 3)\), parallel to \(y = 2x + 1\): \(y = 2x - 5\). Perpendicular: slope \(-\dfrac{1}{2}\), so \(y = -\dfrac{1}{2}x + 5\).

Watch out: perpendicular to slope 2 is \(-\dfrac{1}{2}\), not \(-2\) and not \(\dfrac{1}{2}\).

Graphing a linear inequality

1. Graph the boundary line: solid for \(\le, \ge\); dashed for \(\lt, \gt\). 2. Pick a test point not on the line, often \((0, 0)\). 3. True: shade its side. False: shade the other side.

Example: \(y \lt 2x + 1\): dashed line; \(0 \lt 1\) is true, so shade the side with \((0, 0)\).

Systems: what a solution is

A solution of a system is a point that makes every equation true: where the lines cross.

Substitution

The substitution method: solve one equation for a variable, put that expression into the other equation, solve, then find the other variable.

Example: \(y = 2x - 1\) and \(3x + y = 9\): \(3x + (2x - 1) = 9\), so \(x = 2\), \(y = 3\). Solution \((2, 3)\).

Elimination

The elimination method: multiply one or both equations so a variable has opposite coefficients, then add the equations.

Example: \(2x + 3y = 7\) and \(4x - 3y = 5\). Add: \(6x = 12\), so \(x = 2\); then \(3y = 3\), \(y = 1\). Solution \((2, 1)\).

Systems word problems

Two variables, two equations: one counts things, one counts money (or value). With wind or current: \((r + w)t = d\) going with it, \((r - w)t = d\) against it.

Example: 200 tickets, adults $8, kids $5, total $1,360: \(x + y = 200\), \(8x + 5y = 1360\). So \(3x = 360\): 120 adults, 80 kids.

Example: A plane flies 600 miles in 2 hours with the wind and 3 hours against it: \(r + w = 300\), \(r - w = 200\), so \(r = 250\), \(w = 50\) mph.

Three variables

Use two different pairs of equations to eliminate the same variable. Solve the two-variable system you get, then substitute back.

Example: \(x + y + z = 6\), \(x - y + z = 2\), \(2x + y - z = 1\). First minus second: \(2y = 4\), so \(y = 2\). Then \(x + z = 4\) and \(2x - z = -1\) give \(x = 1\), \(z = 3\): \((1, 2, 3)\).