Linear equations
52 flashcards covering Linear equations for the GMAT Quantitative section.
Linear equations are straightforward mathematical expressions that describe a straight-line relationship between variables. At their core, they involve an equation like y = mx + b, where m is the slope (indicating the rate of change) and b is the y-intercept (the starting point). These equations help model real-world situations, such as calculating costs based on quantity or predicting profits from sales, making them essential for understanding basic algebraic relationships.
On the GMAT Quantitative section, linear equations frequently appear in problem-solving and data sufficiency questions, often involving solving for variables, graphing lines, or handling systems of equations. Common traps include algebraic mistakes, like forgetting to multiply through parentheses or misreading word problem contexts, which can lead to incorrect answers. Focus on practicing quick manipulation of equations, interpreting slopes and intercepts accurately, and translating verbal descriptions into equations to handle these efficiently.
A concrete tip: Always verify your solution by plugging values back into the original equation.
Terms (52)
- 01
Linear equation
A linear equation is an algebraic equation that graphs as a straight line, such as ax + b = c, where a and b are constants and x is the variable.
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Slope of a line
The slope measures the steepness of a line and is calculated as the change in y-coordinates divided by the change in x-coordinates between two points on the line.
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Y-intercept
The y-intercept is the point where a line crosses the y-axis, represented as the value of b in the equation y = mx + b.
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X-intercept
The x-intercept is the point where a line crosses the x-axis, found by setting y to zero in the equation and solving for x.
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Slope-intercept form
Slope-intercept form is the equation of a line written as y = mx + b, where m is the slope and b is the y-intercept.
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Standard form
Standard form of a linear equation is Ax + By = C, where A, B, and C are constants, and A and B are not both zero.
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Point-slope form
Point-slope form is the equation of a line written as y - y1 = m(x - x1), where m is the slope and (x1, y1) is a point on the line.
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Solving a linear equation
Solving a linear equation involves isolating the variable on one side by performing inverse operations, such as adding, subtracting, multiplying, or dividing both sides.
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Graphing a linear equation
Graphing a linear equation means plotting points that satisfy the equation and drawing a straight line through them, often using the slope and intercepts.
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Parallel lines
Parallel lines are lines in the same plane that never intersect and have the same slope but different y-intercepts.
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Perpendicular lines
Perpendicular lines are lines that intersect at a right angle, with slopes that are negative reciprocals of each other.
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Systems of linear equations
Systems of linear equations are two or more equations with the same variables, solved to find values that satisfy all equations simultaneously.
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Substitution method
The substitution method solves a system of equations by solving one equation for one variable and substituting that expression into the other equation.
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Elimination method
The elimination method solves a system by adding or subtracting equations to eliminate one variable, allowing the other to be solved directly.
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Consistent system
A consistent system of linear equations has at least one solution, meaning the equations intersect at one or more points.
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Inconsistent system
An inconsistent system of linear equations has no solution, as the lines are parallel and never intersect.
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Dependent system
A dependent system of linear equations has infinitely many solutions, occurring when the equations represent the same line.
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Break-even point
The break-even point is the level of production or sales where total costs equal total revenue, often found by solving a system of linear equations.
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Linear inequality
A linear inequality is an inequality that represents a region on a graph, such as ax + b > c, and includes solutions where the inequality holds true.
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Rate of change
Rate of change in a linear equation is the constant difference in the dependent variable per unit change in the independent variable, equivalent to the slope.
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Horizontal line
A horizontal line has a slope of zero and is represented by an equation of the form y = k, where k is a constant.
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Vertical line
A vertical line has an undefined slope and is represented by an equation of the form x = h, where h is a constant.
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Distributive property
The distributive property allows multiplying a number by a sum inside parentheses, such as a(b + c) = ab + ac, and is essential for simplifying linear equations.
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Combining like terms
Combining like terms means adding or subtracting terms with the same variables, such as 3x + 2x = 5x, to simplify an equation.
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Isolating variables
Isolating variables requires performing operations on both sides of an equation to get the variable alone, ensuring equality is maintained.
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Checking solutions
Checking solutions involves substituting the found values back into the original equation to verify they satisfy it and avoid errors.
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Word problems with linear equations
Word problems with linear equations translate real-world scenarios into equations, such as distance equals rate times time, and solve for unknowns.
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Mixture problems
Mixture problems involve combining solutions or items with different concentrations, solved using linear equations to find ratios or amounts.
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Work-rate problems
Work-rate problems use linear equations to determine how long it takes for individuals or machines working at constant rates to complete a task together.
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Age problems
Age problems set up linear equations based on relationships between ages at different times, such as current age plus years equaling future age.
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Positive slope
A positive slope indicates that the line rises from left to right, meaning as x increases, y increases.
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Negative slope
A negative slope indicates that the line falls from left to right, meaning as x increases, y decreases.
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Zero slope
Zero slope means the line is horizontal, with no change in y for any change in x.
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Undefined slope
Undefined slope occurs in vertical lines, where x is constant and y can vary, making the slope calculation impossible.
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Linear functions
Linear functions are functions of the form f(x) = mx + b, where the graph is a straight line and the rate of change is constant.
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Domain of a linear function
The domain of a linear function is all real numbers, as there are no restrictions on the x-values that can be plugged into the equation.
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Range of a linear function
The range of a linear function is all real numbers, except for horizontal lines where the range is a single value.
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Example: Solve 2x + 3 = 7
To solve 2x + 3 = 7, subtract 3 from both sides to get 2x = 4, then divide by 2 to find x = 2.
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Example: Find equation with slope 2 and point (1, 3)
Using point-slope form, the equation is y - 3 = 2(x - 1), which simplifies to y = 2x + 1.
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Strategy for graphing lines
To graph a line, plot the y-intercept and use the slope to find another point, then draw the line through those points.
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Identifying linear equations
Linear equations are identified by having variables to the first power and no products of variables, unlike quadratic or exponential equations.
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Constant rate
A constant rate in linear equations means the relationship between variables changes at a steady pace, represented by the slope.
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Intercept form
Intercept form of a line is x/a + y/b = 1, where a is the x-intercept and b is the y-intercept.
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Solution to a system
The solution to a system of linear equations is the point or points that satisfy all equations, found at the intersection of the lines.
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Extraneous solutions
Extraneous solutions are values that appear to solve an equation but do not, often from errors in manipulation, though less common in linear equations.
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Applications in business
In business, linear equations model relationships like cost and quantity, such as total cost equals fixed cost plus variable cost per unit.
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Rate, time, distance problems
Rate, time, distance problems use the equation distance = rate × time to set up and solve linear equations for unknowns.
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Addition method for systems
The addition method is another name for the elimination method, where equations are added after multiplying to align coefficients.
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Independent system
An independent system of linear equations has exactly one solution, meaning the lines intersect at a single point.
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Common mistakes in solving
Common mistakes include forgetting to perform operations on both sides or misapplying the distributive property, leading to incorrect solutions.
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Example: Graph y = 3x - 2
To graph y = 3x - 2, plot the y-intercept at (0, -2) and use the slope of 3 to find another point, like (1, 1), then draw the line.
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Example: Solve system: y = 2x + 1 and y = 3x - 1
Set 2x + 1 = 3x - 1, subtract 2x from both sides to get 1 = x - 1, add 1 to both sides to find x = 2, then y = 2(2) + 1 = 5.