SAT Linear Equations and Slope: From Equations to Graphs to Applications

The slope of a line measures its steepness and direction. Slope=rise/run=(y2-y1)/(x2-x1). A positive slope means the line rises from left to right. A negative slope means the line falls from left to right. A slope of zero is a horizontal line, and undefined slope (division by zero) is a vertical line. The slope-intercept form of a line is y=mx+b, where m is the slope and b is the y-intercept (the y-value where the line crosses the y-axis). This form is useful because you can immediately identify the slope and y-intercept without any computation. When writing a line's equation or identifying its slope and y-intercept, always first convert the equation to slope-intercept form by solving for y, which eliminates errors and makes the slope and intercept immediately visible. For example, 2x+3y=6 becomes y=-2/3*x+2, so the slope is -2/3 and the y-intercept is 2.

Parallel lines have the same slope. Perpendicular lines have slopes that are negative reciprocals of each other, meaning if one line has slope m, the perpendicular line has slope -1/m. For a line with slope 2/3, a perpendicular line has slope -3/2. Questions asking whether two lines are parallel or perpendicular test whether you can identify and compare slopes. If you are given two equations, convert both to slope-intercept form to see their slopes and compare. If you are given two points on each line and asked about the relationship, calculate the slope of each line and compare. This straightforward approach works consistently across all problem types involving parallel and perpendicular lines.

You may be asked to write an equation of a line given two points, a point and a slope, a graph, or a description. If you have two points (x1,y1) and (x2,y2), calculate the slope using the slope formula, then use point-slope form: y-y1=m(x-x1). Substitute one of the points and your calculated slope, then simplify to slope-intercept form. If you have a point and slope, substitute directly into point-slope form. If you have a graph, identify two points on the line (the y-intercept is often one), calculate slope, and write the equation. A common shortcut is to use the y-intercept as one point, since it appears directly in slope-intercept form: once you know the slope and y-intercept, the equation is written immediately without point-slope form. If the problem describes a line (like "passes through (2,5) with slope 3"), use point-slope form, then convert to slope-intercept form for a cleaner final answer. Practice all these approaches until you can identify which method is fastest for each scenario.

Standard form of a line is Ax+By=C. You may need to convert between slope-intercept and standard form. To convert y=-2/3*x+5 to standard form, multiply by 3 to clear fractions: 3y=-2x+15, then rearrange: 2x+3y=15. Standard form is sometimes required in the answer, so check the problem statement. Different forms have different uses: slope-intercept form shows slope and y-intercept immediately; point-slope form emphasizes a point on the line; standard form emphasizes integer coefficients. Choose your form based on what the problem asks for or what makes the work clearest.

The distance between two points (x1,y1) and (x2,y2) is d=sqrt((x2-x1)^2+(y2-y1)^2), derived from the Pythagorean theorem. The midpoint is the point halfway between two points: midpoint=((x1+x2)/2, (y1+y2)/2). These formulas appear on the SAT reference sheet, but knowing them by memory is faster. Questions might ask for the distance between two points, the midpoint of a segment, or properties of a geometric figure defined by points and lines. For a line segment with endpoints (1,3) and (5,7), the distance is sqrt((5-1)^2+(7-3)^2)=sqrt(16+16)=sqrt(32)=4sqrt(2). The midpoint is ((1+5)/2, (3+7)/2)=(3,5). If a question asks about a specific type of geometric figure like an isosceles triangle or a rectangle, use distance and midpoint formulas to verify the properties (equal side lengths, right angles) rather than trying to visualize the figure. This algebraic approach is reliable and avoids the errors that come from sketching figures by hand.

Some problems define a line or a figure algebraically and ask you to identify points that satisfy conditions. For instance, "all points equidistant from (0,0) and (4,0)" describes the perpendicular bisector of the segment from (0,0) to (4,0). The perpendicular bisector passes through the midpoint (2,0) and is perpendicular to the original segment. If the original segment is horizontal, the perpendicular bisector is vertical: x=2. These geometric insights combine with algebraic formulas to solve problems efficiently. Practice translating geometric conditions into equations and vice versa, as this skill is essential for coordinate geometry problems.

Word problems often involve linear relationships. A car traveling at 60 mph for t hours covers a distance of d=60t. The y-intercept (0,0) means at time zero, distance is zero. The slope of 60 represents the rate of travel. When graphing, identify the domain and range based on the context. Time cannot be negative, and distance cannot be negative, so you graph only in the first quadrant. Some linear models have a starting value that is not zero, like "a subscription costs $10 initially plus $5 per month." This becomes y=5x+10, where x is months and y is total cost. The y-intercept of 10 represents the initial cost. Understanding how slope and y-intercept translate to real-world quantities helps you verify that your linear equation makes sense in context and that you have not mixed up slope and intercept. After writing an equation, test it by substituting a realistic point: does x=2 months give a reasonable cost? Does increasing x increase the cost appropriately?

Graphing lines in a coordinate plane requires identifying at least two points, plotting them, and drawing a line through them. Using the y-intercept and one additional point using the slope is efficient. For y=-2x+3, the y-intercept is (0,3). Using slope -2, move right 1 and down 2 to get another point (1,1). Draw a line through these two points. For more complex lines or when you want to verify your graph, use Desmos to plot the equation and check that it looks reasonable. On test day, if a question presents a graph and asks you to identify the equation, use two points from the graph to find the slope, then read the y-intercept from the graph where the line crosses the y-axis.