Linear equations
59 flashcards covering Linear equations for the ACT Math section.
Linear equations are straightforward mathematical tools that describe a straight-line relationship between variables, such as x and y. For instance, an equation like y = 2x + 3 shows how y changes as x increases, with a constant rate of change. They form the foundation of algebra, helping to model real-world scenarios like budgeting or motion, and they're essential for building skills in more complex math.
On the ACT Math section, linear equations show up in questions about solving for variables, graphing lines, interpreting slopes and y-intercepts, and handling systems of equations. Common traps include algebraic errors, like forgetting to multiply through parentheses, or misreading word problems that disguise equations in everyday contexts. Focus on practicing quick solving techniques, recognizing parallel and perpendicular lines, and applying equations to scenarios like distance or cost.
Double-check your work by substituting solutions back into the original equation.
Terms (59)
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Linear equation
A linear equation is an equation that represents a straight line when graphed, typically in the form of 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 any two points on the line.
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Y-intercept
The y-intercept is the point where a line crosses the y-axis, representing the value of y when x is zero in the equation of the line.
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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 written as 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 used to write the equation of a line as y - y1 = m(x - x1), where m is the slope and (x1, y1) is a point on the line.
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X-intercept
The x-intercept is the point where a line crosses the x-axis, representing the value of x when y is zero.
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Graphing a linear equation
Graphing a linear equation involves plotting points that satisfy the equation and drawing a straight line through them to visualize the relationship.
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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, and their slopes are negative reciprocals of each other.
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Systems of linear equations
Systems of linear equations consist of two or more equations with the same variables, and solving them means finding 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 you to solve for the remaining variable.
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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, occurring when the lines are parallel and never intersect.
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Dependent system
A dependent system of linear equations has infinitely many solutions, happening when the equations represent the same line.
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Solving linear equations
Solving linear equations involves isolating the variable on one side by performing inverse operations, such as adding, subtracting, multiplying, or dividing both sides.
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Linear inequality
A linear inequality is similar to a linear equation but uses symbols like <, >, ≤, or ≥, representing a range of values rather than a single solution.
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Graphing linear inequalities
Graphing linear inequalities involves graphing the boundary line and shading the region that satisfies the inequality, using a solid or dashed line based on the symbol.
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Boundary line
The boundary line in a linear inequality is the line that separates the regions where the inequality is true from where it is false.
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Rate of change
Rate of change in a linear equation is the constant ratio of change in the dependent variable to the change in the independent variable, equivalent to the slope.
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Direct variation
Direct variation describes a linear relationship where one variable is a constant multiple of the other, expressed as y = kx, where k is the constant of variation.
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Horizontal line
A horizontal line has a slope of zero and is represented by an equation of the form y = c, where c 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 = c, where c is a constant.
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Undefined slope
Undefined slope occurs in vertical lines, where the change in x is zero, making the slope calculation impossible.
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Zero slope
Zero slope occurs in horizontal lines, where the change in y is zero, indicating no vertical change.
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Equation from two points
To find the equation of a line from two points, first calculate the slope, then use one point and the point-slope form to write the equation.
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Intercepts method
The intercepts method graphs a line by finding the x-intercept and y-intercept and connecting those points with a straight line.
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Word problems with linear equations
Word problems with linear equations require translating real-world scenarios into equations and solving for unknowns, such as distance, rate, and time.
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Distance-rate-time problems
In distance-rate-time problems, the equation distance = rate × time is used to set up and solve linear equations for unknown variables.
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Mixture problems
Mixture problems involve combining solutions or items with different concentrations, leading to linear equations to find amounts or concentrations.
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Break-even point
The break-even point is the level of production or sales where total costs equal total revenue, found by solving a system of linear equations.
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Common trap: dividing by zero
Dividing by zero is a common error when solving equations, as it leads to undefined results and invalid solutions.
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Common trap: sign errors
Sign errors occur when incorrectly handling negative signs during addition, subtraction, or distribution in linear equations.
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Common trap: misreading equations
Misreading equations, such as confusing coefficients or constants, can lead to incorrect setups and solutions in linear problems.
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Linear functions
Linear functions are functions whose graph is a straight line, generally expressed as f(x) = mx + b, where m is the slope and b is the y-intercept.
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Domain of linear functions
The domain of a linear function is all real numbers, as there are no restrictions on the input values for a straight line.
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Range of linear functions
The range of a linear function is all real numbers, except in the case of horizontal lines, where it is a single value.
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Increasing linear function
An increasing linear function has a positive slope, meaning as x increases, y also increases.
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Decreasing linear function
A decreasing linear function has a negative slope, meaning as x increases, y decreases.
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Positive slope
A positive slope indicates that the line rises from left to right, showing a direct relationship between x and y.
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Negative slope
A negative slope indicates that the line falls from left to right, showing an inverse relationship between x and y.
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Using tables to graph lines
Using tables to graph lines involves creating a table of values by plugging in x-values into the equation and plotting the corresponding y-values.
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Solving equations with fractions
Solving linear equations with fractions requires multiplying through by the least common denominator to eliminate fractions before isolating the variable.
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Solving equations with decimals
Solving linear equations with decimals often involves multiplying through by a power of ten to convert decimals to whole numbers for easier calculation.
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Extraneous solutions in equations
Although rare in linear equations, extraneous solutions can arise from algebraic manipulations, so always verify solutions by substitution.
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Cost and revenue equations
Cost and revenue equations model business scenarios, where total cost and revenue are linear functions, and their intersection is the break-even point.
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Supply and demand equations
Supply and demand equations are linear and used to find equilibrium price and quantity by solving the system where supply equals demand.
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Arithmetic sequences
Arithmetic sequences are linear patterns where each term increases by a constant difference, represented by the formula an = a1 + (n-1)d.
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Midpoint of a line segment
The midpoint of a line segment is the point halfway between two endpoints, calculated as ((x1 + x2)/2, (y1 + y2)/2).
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Slope from a graph
To find the slope from a graph, select two points on the line and calculate the rise over run between them.
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Converting to slope-intercept form
Converting an equation to slope-intercept form involves solving for y to identify the slope and y-intercept easily.
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Parallel lines slopes
Lines are parallel if they have the same slope, which is key for determining if equations represent parallel lines.
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Perpendicular lines slopes
Lines are perpendicular if the product of their slopes is -1, provided neither is vertical.
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Linear equation in one variable
A linear equation in one variable, like 2x + 3 = 7, has a single solution and graphs as a point on the number line.
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Linear equation in two variables
A linear equation in two variables, like 2x + y = 5, represents a line in the coordinate plane.
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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.
This equation has one solution: x = 2.
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Example: Graph y = 2x + 1
To graph y = 2x + 1, plot the y-intercept at (0,1) and use the slope of 2 to find another point, like (1,3), then draw the line.
The line passes through (0,1) and (1,3).
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Example: Find equation from points (1,2) and (3,6)
For points (1,2) and (3,6), calculate slope m = (6-2)/(3-1) = 2, then use point-slope form: y - 2 = 2(x - 1), simplifying to y = 2x.