Linear Relationships Cheat Sheet

Mastering linear relationships involves understanding their equations, graphing them accurately, and identifying key properties like intercepts, gradients, and parallel/perpendicular relationships.

Core Principles

The equation of a line in the form y = mx + c, where 'm' is the gradient and 'c' is the y-intercept.
To determine if a point lies on a line, substitute its coordinates into the line's equation; if the equation holds true, the point is on the line.
Lines are parallel if they have the same gradient; lines are perpendicular if the product of their gradients is -1.
The x-intercept is where the line crosses the x-axis (y=0), and the y-intercept is where the line crosses the y-axis (x=0).

Action Steps

To find if a point (-2,3) lies on y = x + 4, substitute x=-2 and y=3: 3 = -2 + 4, which is 3 = 2 (False).
To find if a point (-2,3) lies on y = -2x - 1, substitute x=-2 and y=3: 3 = -2(-2) - 1, which is 3 = 4 - 1, so 3 = 3 (True).
To sketch 2x + 4y = 8, find x-intercept (y=0): 2x = 8, x=4. Find y-intercept (x=0): 4y = 8, y=2. Plot (4,0) and (0,2).
To sketch y - 2x = -4, find x-intercept (y=0): -2x = -4, x=2. Find y-intercept (x=0): y = -4. Plot (2,0) and (0,-4).
To find the equation of a horizontal line passing through y=3, the equation is y = 3.
To find the equation of a vertical line passing through x=-4, the equation is x = -4.
To sketch a line parallel to y = -2x + 3, choose a different y-intercept, e.g., y = -2x + 1.
To sketch a line perpendicular to y = -2x + 3, find the negative reciprocal of the gradient (-2), which is 1/2. Choose a y-intercept, e.g., y = 1/2x + 1.
To solve 18 = 3x + 6, subtract 6 from both sides: 12 = 3x, then divide by 3: x = 4.
To solve (2x-3)/5 = 3, multiply both sides by 5: 2x - 3 = 15, add 3: 2x = 18, divide by 2: x = 9.
To find the gradient of y = 2x + 5, identify 'm' in y = mx + c, so the gradient is 2.
To find the gradient between A(1,3) and B(3,4), use the formula m = (y2 - y1) / (x2 - x1) = (4-3) / (3-1) = 1/2.
To state the y-intercept of the graphed line, find where it crosses the y-axis. For the line shown, it is -1.
To calculate the gradient of the graphed line, choose two points, e.g., (0,-1) and (3,-5). m = (-5 - (-1)) / (3 - 0) = -4 / 3.
To state the equation of the line from Question 5b, use y = mx + c with m = -4/3 and c = -1: y = -4/3x - 1.
To write the equation of a line with gradient m=2 passing through (1,-3), use y - y1 = m(x - x1): y - (-3) = 2(x - 1) => y + 3 = 2x - 2 => y = 2x - 5.
To find the equation of the line passing through (0,-3) and (4,0), first find the gradient: m = (0 - (-3)) / (4 - 0) = 3/4. Use y-intercept (0,-3): y = 3/4x - 3.
To rearrange 6x - 2y = 10 into y = mx + c, isolate y: -2y = -6x + 10, divide by -2: y = 3x - 5.
From y = 3x - 5, the gradient is 3 and the y-intercept is -5.
To solve 5x + 4 ≥ 29, subtract 4: 5x ≥ 25, divide by 5: x ≥ 5.
To solve 1 - x/5 < -2, subtract 1: -x/5 < -3, multiply by -5 and reverse inequality: x > 15.
To solve 4x - 5 ≥ 7x - 3, subtract 4x: -5 ≥ 3x - 3, add 3: -2 ≥ 3x, divide by 3: -2/3 ≥ x, or x ≤ -2/3.
To solve 4x - 3 = 2x + 15, subtract 2x: 2x - 3 = 15, add 3: 2x = 18, divide by 2: x = 9.
To solve 3(x - 2) + 2x = 14, distribute: 3x - 6 + 2x = 14, combine x terms: 5x - 6 = 14, add 6: 5x = 20, divide by 5: x = 4.
To solve (2x+3)/4 = (x-2)/3, cross-multiply: 3(2x+3) = 4(x-2), distribute: 6x + 9 = 4x - 8, subtract 4x: 2x + 9 = -8, subtract 9: 2x = -17, divide by 2: x = -17/2.
To complete the table for the theme park cost: 0 rides = $12, 1 ride = $16, 2 rides = $20, 3 rides = $24.
To write the equation for cost C and rides n, use C = 4n + 12.
To find how many rides Juan can afford with $40, solve 40 = 4n + 12: 28 = 4n, n = 7 rides.
To find the equation of line B perpendicular to y = 2x + 1 and passing through (4,1), the gradient of B is -1/2. Use point-slope form: y - 1 = -1/2(x - 4) => y - 1 = -1/2x + 2 => y = -1/2x + 3.
To determine if 4x - 2y = -6 is parallel to y = 2x + 1, rearrange 4x - 2y = -6 to y = mx + c: -2y = -4x - 6 => y = 2x + 3. Since both lines have a gradient of 2, they are parallel.

Key Terms

Gradient (m): The steepness of a line, calculated as the change in y divided by the change in x (rise over run).
Y-intercept (c): The point where a line crosses the y-axis, occurring when x = 0.
X-intercept: The point where a line crosses the x-axis, occurring when y = 0.
Parallel lines: Lines that have the same gradient and never intersect.
Perpendicular lines: Lines that intersect at a right (90-degree) angle; the product of their gradients is -1.

Real World Examples

A theme park charges a $12 entry fee plus $4 per ride.: This can be modeled by the linear equation C = 4n + 12, where C is the total cost and n is the number of rides. This allows calculation of costs for any number of rides and determination of how many rides can be afforded within a budget.