Linear equations are the building blocks of algebra. They help us model real-world relationships and solve practical problems. Understanding how to manipulate these equations is crucial for tackling more complex math concepts.
In this section, we'll learn to solve linear equations, work with fractions, and construct equations from given information. We'll also explore relationships between lines, including and lines, which are essential for geometry and graphing.
Solving Linear Equations and Analyzing Line Relationships
Solving linear equations
Isolate the variable term on one side of the equation by adding or subtracting terms from both sides to move non-variable terms to the opposite side and multiplying or dividing both sides by the same value
Simplify the equation by combining like terms (3x + 2x = 5x)
Solve for the variable by performing the inverse operation on both sides such as addition if the variable is being subtracted (x - 5 = 10, add 5 to both sides) or division if the variable is being multiplied (2x = 10, divide both sides by 2)
The set of all values that satisfy the equation is called the
Rational equations with variables
Identify rational equations containing fractions with variables ((x+1)/(x−2)=3)
Find the (LCD) of all fractions in the equation and multiply both sides by the LCD to eliminate fractions
Simplify the equation by combining like terms then solve for the variable using algebraic methods
Check the solution by substituting it back into the original equation and verify it does not result in a zero denominator
Construction of linear equations
Identify the slope and y-intercept from given information where slope can be determined from two points ((2,3) and (4,7)) or a point and a direction and y-intercept is the point where the line crosses the y-axis
Use the slope-intercept form of a y=mx+b where m represents the slope and b represents the y-intercept
Substitute the slope and y-intercept values into the equation and simplify if necessary
Relationships between lines
Parallel lines have the same slope but different y-intercepts so the equation of a line parallel to y=mx+b is y=mx+c where c is a different y-intercept
Perpendicular lines have slopes that are negative reciprocals of each other so if the slope of line 1 is m1, the slope of line 2 perpendicular to line 1 is m2=−1/m1
Vertical lines are perpendicular to horizontal lines as vertical lines have undefined slope and equation x=a while horizontal lines have a slope of zero and equation y=b
Parallel and perpendicular line equations
Given the equation of a line, identify its slope and y-intercept
For a parallel line, use the same slope and a different y-intercept then substitute into the slope-intercept form y=mx+b
For a perpendicular line:
Find the negative of the given line's slope
Use the new slope and a given point to find the y-intercept
Substitute the new slope and y-intercept into the slope-intercept form y=mx+b
Additional Concepts
Algebraic expressions: Mathematical phrases that can include variables, constants, and operations
Inequalities: Mathematical statements that compare two expressions using inequality symbols (<, >, ≤, ≥)
Domain: The set of all possible input values (x-values) for a function or relation
Range: The set of all possible output values (y-values) for a function or relation
Key Terms to Review (14)
Horizontal line: A horizontal line is a straight line that runs left to right and has a constant y-value for all points on the line. In a Cartesian coordinate system, its slope is zero.
Identity equation: An identity equation is an equation that holds true for all values of the variable. In other words, both sides of the equation are equivalent expressions.
Inconsistent equation: An inconsistent equation is a linear equation in one variable that has no solution. This occurs when the equation simplifies to a contradiction, such as $0 = 1$.
Least common denominator: The least common denominator (LCD) is the smallest positive number that is a multiple of the denominators of two or more fractions. It is used to add, subtract, or compare fractions.
Linear equation: A linear equation in one variable is an equation that can be expressed in the form $ax + b = 0$, where $a$ and $b$ are constants, and $x$ is the variable. The solution to a linear equation in one variable is the value of $x$ that makes the equation true.
Parallel: Parallel lines are two or more lines in a plane that never intersect. They have the same slope but different y-intercepts.
Perpendicular: Perpendicular lines are two lines that intersect at a right angle (90 degrees). In the context of linear equations, their slopes are negative reciprocals of each other.
Point-slope formula: The point-slope formula is a method for finding the equation of a line when you know the slope and one point on the line. It is written as $y - y_1 = m(x - x_1)$, where $(x_1, y_1)$ is a given point and $m$ is the slope.
Rational expression: A rational expression is a fraction where both the numerator and the denominator are polynomials. It is defined for all values of the variable except those that make the denominator zero.
Rational number: A rational number is any number that can be expressed as the quotient or fraction of two integers, where the numerator is an integer and the denominator is a non-zero integer. Rational numbers include integers, fractions, and finite or repeating decimals.
Reciprocal: The reciprocal of a number is 1 divided by that number. For a non-zero number $a$, its reciprocal is $\frac{1}{a}$.
Solution set: A solution set is the collection of all possible solutions that satisfy a given equation or system of equations. It represents the values of variables that make the equation true.
Standard form: Standard form in the context of linear equations is a way of writing equations in the format $Ax + B = C$, where $A$, $B$, and $C$ are constants and $x$ is the variable. This form helps to easily identify and manipulate the components of an equation.
Vertical line: A vertical line is a straight line that goes up and down and has an undefined slope. It is represented by an equation of the form $x = a$, where $a$ is a constant.