Linear equations
49 flashcards covering Linear equations for the SAT Math section.
Linear equations are straightforward mathematical tools that describe straight-line relationships between variables. At their core, they take the form y = mx + b, where m represents the slope (how steep the line is) and b is the y-intercept (where the line crosses the y-axis). For someone new to the concept, think of them as equations that show how one quantity changes predictably with another, like the cost of a taxi ride based on distance traveled. They’re essential for understanding basic patterns in algebra and real-world applications, such as budgeting or predicting trends.
On the SAT Math section, linear equations appear in multiple-choice and grid-in questions, often involving solving for variables, graphing lines, or interpreting word problems that translate scenarios into equations. Common traps include algebraic errors like distributing negatives incorrectly or overlooking restrictions on variables, so watch for misleading answer choices that exploit these mistakes. Focus on mastering skills like isolating variables, understanding slope and intercepts, and converting real-world situations into equations to handle these problems efficiently.
For a quick tip: Practice substituting values back into the equation to double-check your work.
Terms (49)
- 01
Linear equation
An equation that graphs as a straight line, such as ax + b = 0 for one variable or ax + by = c for two variables, where a, b, and c are constants and a and b are not both zero.
- 02
Slope of a line
The measure of a line's steepness, calculated as the ratio of the vertical change to the horizontal change between any two points on the line, often denoted as m in y = mx + b.
- 03
Y-intercept
The point where a line crosses the y-axis, representing the value of y when x is 0, commonly found in the equation y = mx + b as the constant b.
- 04
X-intercept
The point where a line crosses the x-axis, representing the value of x when y is 0, found by setting y to 0 in the equation and solving for x.
- 05
Slope-intercept form
A way to write the equation of a line as y = mx + b, where m is the slope and b is the y-intercept, making it easy to graph and identify key features.
- 06
Point-slope form
A form of a line's equation given as y - y1 = m(x - x1), where m is the slope and (x1, y1) is a point on the line, useful for writing equations from a point and slope.
- 07
Standard form of a line
An equation written as ax + by = c, where a, b, and c are constants, allowing for easy identification of intercepts and use in certain algebraic methods.
- 08
Parallel lines
Lines in the same plane that never intersect and have the same slope, meaning their equations have identical m values in slope-intercept form.
- 09
Perpendicular lines
Lines that intersect at a 90-degree angle, with slopes that are negative reciprocals of each other, such as m1 m2 = -1.
- 10
Solving a one-variable linear equation
The process of isolating the variable to find its value, using operations like addition, subtraction, multiplication, and division while maintaining equality on both sides.
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System of linear equations
A set of two or more linear equations solved simultaneously to find values that satisfy all equations at once, often representing intersecting lines.
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Substitution method
A technique for solving systems of equations by solving one equation for one variable and substituting that expression into the other equation to solve for the remaining variable.
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Elimination method
A method for solving systems of equations by adding or subtracting the equations to eliminate one variable, allowing you to solve for the other variable directly.
- 14
Graphing a linear equation
The process of plotting points that satisfy the equation and drawing a straight line through them, using intercepts or slope to ensure accuracy.
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Horizontal line
A line that runs parallel to the x-axis and has a constant y-value, represented by an equation of the form y = k, where k is a constant, and its slope is zero.
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Vertical line
A line that runs parallel to the y-axis and has a constant x-value, represented by an equation of the form x = h, where h is a constant, and its slope is undefined.
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Undefined slope
The slope of a vertical line, which occurs when the change in x is zero, making the slope calculation impossible as division by zero.
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Zero slope
The slope of a horizontal line, which is zero because there is no change in y-values, indicating the line is flat.
- 19
Linear inequality
An inequality involving a linear expression, such as ax + b > c, where the solution is a range of values rather than a single point, often graphed as a shaded region.
- 20
Graphing linear inequalities
The process of graphing the corresponding linear equation and then shading the region that satisfies the inequality, using a dashed or solid line based on the inequality symbol.
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Solution set
The set of all values that satisfy a linear equation or inequality, such as all points on a line for an equation or a half-plane for an inequality.
- 22
Break-even point
In word problems, the point where costs equal revenues, found by solving a system of equations representing the two quantities.
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Distance formula
A formula to calculate the distance between two points (x1, y1) and (x2, y2) as sqrt((x2 - x1)^2 + (y2 - y1)^2), often used in problems involving lines.
- 24
Midpoint formula
A formula to find the midpoint of a line segment between two points (x1, y1) and (x2, y2) as ((x1 + x2)/2, (y1 + y2)/2), helpful for analyzing lines.
- 25
Rate of change
The slope of a line in a real-world context, representing how one quantity changes with respect to another, such as speed as distance over time.
- 26
Direct variation
A relationship where one variable is a constant multiple of another, expressed as y = kx, which is a linear equation passing through the origin.
- 27
Mixture problems
Word problems involving combining solutions or items with different concentrations, solved by setting up and solving a linear equation for the unknown quantity.
- 28
Work rate problems
Problems where entities work at constant rates, solved by setting up equations based on the formula work = rate × time to find unknown times or rates.
- 29
Distance, speed, and time problems
Word problems using the formula distance = speed × time, often requiring a linear equation to relate variables like speed and time.
- 30
Dividing by zero error
A common mistake in solving linear equations where division by a variable that could be zero leads to incorrect solutions, which must be checked.
- 31
Absolute value equation
An equation like |ax + b| = c, which can be solved by considering two cases: ax + b = c and ax + b = -c, resulting in linear equations.
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Linear function
A function whose graph is a straight line, defined by an equation like f(x) = mx + b, where the output changes at a constant rate with the input.
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Domain of a linear function
The set of all possible input values for a linear function, which is typically all real numbers unless restricted by context.
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Range of a linear function
The set of all possible output values for a linear function, which is also all real numbers for non-horizontal lines.
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Increasing linear function
A linear function with a positive slope, meaning as x increases, y increases at a constant rate.
- 36
Decreasing linear function
A linear function with a negative slope, meaning as x increases, y decreases at a constant rate.
- 37
Intercepts method
A graphing technique that involves finding the x-intercept and y-intercept of a line and connecting them to draw the graph quickly.
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Slope as rise over run
A way to interpret slope as the ratio of vertical change (rise) to horizontal change (run) between two points, such as 3/4 meaning 3 units up for every 4 units right.
- 39
Equation from two points
The process of finding a line's equation by first calculating the slope from two given points, then using one point to find the y-intercept or point-slope form.
- 40
Equation from point and slope
The method of writing a line's equation using a given point and slope, typically by plugging into point-slope form and simplifying.
- 41
Parallel lines equation
Given one line's equation, the equation of a parallel line can be written by using the same slope and a different y-intercept.
- 42
Perpendicular lines equation
Given one line's slope, the equation of a perpendicular line uses the negative reciprocal slope and passes through a specified point.
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System with no solution
A system of linear equations that represents parallel lines, meaning the equations are inconsistent and have no common solution.
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System with infinite solutions
A system where the equations represent the same line, so every point on the line is a solution.
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Inconsistent system
A system of equations with no solution, occurring when the lines are parallel and distinct.
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Dependent system
A system where the equations are equivalent, representing the same line and thus having infinite solutions.
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Independent system
A system with exactly one solution, occurring when the lines intersect at a single point.
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Linear equation word problem setup
The step of translating a word problem into a linear equation by defining variables for unknowns and using relationships described in the problem.
- 49
Checking solutions
The practice of substituting solved values back into the original equation to verify they satisfy it, catching any algebraic errors.