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2.2 Linear Equations in One Variable

4 min read•june 24, 2024

are the building blocks of algebra, helping us model real-world relationships. They're like simple recipes, mixing variables and constants to create mathematical statements. Understanding how to solve them is crucial for tackling more complex problems in math and science.

Mastering linear equations opens doors to analyzing trends, making predictions, and solving practical problems. From calculating slopes to finding intersections, these skills form the foundation for advanced math concepts. They're the first step in translating real-life scenarios into solvable mathematical expressions.

Solving Linear Equations and Understanding Lines

Key Components of Linear Equations

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  • : The unknown quantity in an equation, typically represented by a letter (e.g., x, y)
  • : The numerical factor of a variable (in 3x + 2, 3 is the coefficient of x)
  • : A term in the equation with a fixed value, not containing a variable (in 3x + 2 = 7, 2 and 7 are constants)

Solving linear equations

  • Isolate the variable on one side of the equation by using to move terms to the other side
    • Add or subtract the same value from both sides to move terms without variables (5x + 3 = 8 becomes 5x = 5)
    • Multiply or divide both sides by the same non-zero value to move terms with variables (2x = 10 becomes x = 5)
  • Simplify the equation until the variable is alone on one side by combining and distributing multiplication over addition or subtraction (3(x + 2) = 9 becomes 3x + 6 = 9)
  • Check the by substituting the value back into the original equation to ensure it holds true (if x = 1, then 3(1 + 2) = 9 is true)
  • The value that satisfies the equation is called the solution

Rational equations and extraneous solutions

  • Find the of all terms in the equation to simplify the equation (2x+1+3x−1=1\frac{2}{x+1} + \frac{3}{x-1} = 1 has an LCD of (x+1)(x−1)(x+1)(x-1))
  • Multiply both sides of the equation by the LCD to clear the denominators, resulting in a linear equation (2x+1+3x−1=1\frac{2}{x+1} + \frac{3}{x-1} = 1 becomes 2(x−1)+3(x+1)=(x+1)(x−1)2(x-1) + 3(x+1) = (x+1)(x-1))
  • Solve the resulting linear equation using algebraic methods to find potential solutions (2x−2+3x+3=x2−12x - 2 + 3x + 3 = x^2 - 1 becomes 5x+1=x2−15x + 1 = x^2 - 1, which simplifies to x2−5x−2=0x^2 - 5x - 2 = 0)
  • Check the solutions in the original to identify , which satisfy the simplified equation but not the original one (x = -1 is an extraneous solution because it makes the original equation undefined)

Slope and y-intercept interpretation

  • (mm) represents the rate of change or steepness of the line, calculated as m=y2−y1x2−x1m = \frac{y_2 - y_1}{x_2 - x_1} for any two points (x1,y1)(x_1, y_1) and (x2,y2)(x_2, y_2) on the line
    • A positive indicates an increasing line (as x increases, y increases), while a negative slope indicates a decreasing line (as x increases, y decreases)
    • A slope of 0 represents a , and an undefined slope represents a
  • (bb) is the point where the line crosses the y-axis, found by setting x=0x = 0 in the equation and solving for yy (in the equation y=2x+3y = 2x + 3, the y-intercept is 3)
  • The of a linear equation is y=mx+by = mx + b, where mm is the slope and bb is the y-intercept

Parallel vs perpendicular lines

  • have the same slope but different y-intercepts
    • For lines y=m1x+b1y = m_1x + b_1 and y=m2x+b2y = m_2x + b_2, if m1=m2m_1 = m_2, the lines are (the lines y=2x+1y = 2x + 1 and y=2x−3y = 2x - 3 are parallel)
  • have slopes that are of each other
    • For lines y=m1x+b1y = m_1x + b_1 and y=m2x+b2y = m_2x + b_2, if m1⋅m2=−1m_1 \cdot m_2 = -1, the lines are (the lines y=2x+1y = 2x + 1 and y=−12x+3y = -\frac{1}{2}x + 3 are perpendicular)

Constructing parallel and perpendicular equations

  • To find an equation parallel to a given line y=mx+by = mx + b:
    1. Use the same slope mm
    2. Choose a different y-intercept b′b'
    3. The parallel line equation is y=mx+b′y = mx + b' (an equation parallel to y=2x+1y = 2x + 1 could be y=2x−3y = 2x - 3)
  • To find an equation perpendicular to a given line y=mx+by = mx + b:
    1. Find the negative reciprocal of the slope: m′=−1mm' = -\frac{1}{m} (for y=2x+1y = 2x + 1, m′=−12m' = -\frac{1}{2})
    2. Choose a point (x1,y1)(x_1, y_1) that the perpendicular line should pass through (let's choose (1, 3))
    3. Use the to find the equation: y−y1=m′(x−x1)y - y_1 = m'(x - x_1) (substituting values, we get y−3=−12(x−1)y - 3 = -\frac{1}{2}(x - 1))
    4. Simplify the equation to y=m′x+b′y = m'x + b' (the perpendicular line equation is y=−12x+72y = -\frac{1}{2}x + \frac{7}{2})

Equation Solving Process

  • Start with an , which is a statement that two expressions are equal
  • Use algebraic operations to isolate the variable on one side of the equation
  • The goal of is to find the value of the variable that makes the equality true

Key Terms to Review (40)

Binomial coefficient: A binomial coefficient is a coefficient of any of the terms in the expansion of a binomial raised to a power, typically written as $\binom{n}{k}$ or $C(n,k)$. It represents the number of ways to choose $k$ elements from a set of $n$ elements without regard to order.
Coefficient: A coefficient is a numerical factor that multiplies a variable in an algebraic expression. It represents the magnitude or strength of the relationship between the variable and the overall expression. Coefficients are essential in various mathematical contexts, including polynomial factorization, linear equations, quadratic equations, and the graphing of polynomial functions.
Conditional equation: A conditional equation is an equation that is true for certain values of the variable(s) involved but not for all values. It typically has specific solutions that satisfy the equation.
Constant: A constant is a value that does not change. In algebra, it is often a number without any variables attached to it.
Constant: A constant is a fixed value that does not change within a given context or problem. It is a fundamental quantity that remains the same regardless of the circumstances or variables involved.
Dependent variable: The dependent variable is the output of a function, whose value depends on the input or independent variable. It is usually represented as $y$ in the equation $y = f(x)$.
Distributive property: The distributive property states that multiplying a sum by a number gives the same result as multiplying each addend by the number and then adding the products. It is expressed as $a(b + c) = ab + ac$.
Distributive Property: The distributive property is a fundamental algebraic rule that allows for the simplification of expressions involving multiplication and addition or subtraction. It states that the product of a number and a sum is equal to the sum of the products of the number with each addend.
Equality: Equality is a fundamental concept that describes the state of being equal, where individuals or objects have the same value, status, rights, or opportunities. It is a central idea in mathematics, particularly in the context of linear equations, where the equality sign represents the balance or equivalence between the expressions on both sides of the equation.
Equation Solving: Equation solving is the process of finding the value(s) of the unknown variable(s) in an equation by applying various mathematical operations and techniques. It is a fundamental skill in mathematics that allows us to solve problems and find solutions to real-world situations.
Extraneous solutions: Extraneous solutions are solutions derived from the algebraic manipulation of an equation that do not satisfy the original equation. They often arise when both sides of an equation are squared or when absolute value functions are involved.
Extraneous Solutions: Extraneous solutions are solutions to an equation that do not satisfy the original conditions or restrictions of the problem. They are solutions that are introduced during the process of solving the equation but do not belong to the set of valid solutions for the given problem.
Horizontal line: A horizontal line is a straight line that runs left to right and has a constant y-value for all points. Its slope is zero because there is no vertical change as you move along the line.
Inconsistent equation: An inconsistent equation is a linear equation with no solution. It represents lines that are parallel and never intersect.
Inverse Operations: Inverse operations are mathematical operations that undo or reverse the effects of another operation. They are used to solve equations by isolating the unknown variable on one side of the equation.
Isolating the Variable: Isolating the variable is the process of rearranging an equation to solve for a specific variable by moving all other variables and constants to one side of the equation. This technique is essential for solving linear equations in one variable, as it allows you to determine the value of the unknown variable.
Least common denominator: The least common denominator (LCD) of two or more rational expressions is the smallest positive integer that is divisible by each of their denominators. It is essential for adding, subtracting, or comparing fractions.
Least Common Denominator (LCD): The least common denominator (LCD) is the smallest positive integer that is divisible by all the denominators in a set of fractions. It is a crucial concept in mathematics, particularly in the context of operations involving fractions, such as adding, subtracting, multiplying, and dividing.
Like Terms: Like terms are algebraic expressions that have the same variable(s) raised to the same power. They can be combined by adding or subtracting their coefficients, as they represent the same type of quantity.
Linear Equations: A linear equation is a mathematical equation in which the variables are raised only to the first power and the equation forms a straight line when graphed. These equations are fundamental in algebra and have numerous applications in various fields.
Negative Reciprocals: Negative reciprocals refer to the reciprocal of a negative number. The reciprocal of a number is the value obtained by dividing 1 by that number. When the original number is negative, the reciprocal will also be negative, creating a negative reciprocal.
Parallel: Parallel lines are lines in a plane that never intersect. They have the same slope but different y-intercepts.
Parallel Lines: Parallel lines are two or more lines that lie in the same plane and never intersect, maintaining a constant distance between them. This concept is fundamental in understanding linear equations and functions, as parallel lines share important geometric properties.
Perpendicular: Perpendicular lines are two lines that intersect at a right angle (90 degrees). In the Cartesian plane, their slopes are negative reciprocals of each other.
Perpendicular lines: Perpendicular lines are lines that intersect at a right angle, which is 90 degrees. Their slopes are negative reciprocals of each other in the coordinate plane.
Perpendicular Lines: Perpendicular lines are a pair of lines that intersect at a right angle, forming a 90-degree angle between them. This geometric relationship is an important concept in the study of linear equations and functions.
Point-Slope Form: The point-slope form is an equation that represents a linear function by specifying a point on the line and the slope of the line. It is a useful way to write the equation of a line when you know a point it passes through and the slope of the line.
Point-slope formula: The point-slope formula is a method used to find the equation of a line given a point on the line and its slope. It is expressed as $y - y_1 = m(x - x_1)$, where $(x_1, y_1)$ is a point on the line and $m$ is the slope.
Rational equation: A rational equation is an equation that involves at least one rational expression, which is a fraction with a polynomial in both the numerator and the denominator. Solving these equations typically involves finding a common denominator or multiplying through by the least common denominator to clear fractions.
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.
Slope: Slope measures the steepness and direction of a line, typically defined as the ratio of the vertical change to the horizontal change between two points on the line. It is commonly represented by the letter $m$.
Slope: Slope is a measure of the steepness or incline of a line or a surface. It represents the rate of change between two variables, typically the change in the vertical direction (y-coordinate) with respect to the change in the horizontal direction (x-coordinate).
Slope-intercept form: Slope-intercept form is a way to express the equation of a straight line using the formula $y = mx + b$. In this formula, $m$ represents the slope and $b$ represents the y-intercept.
Slope-Intercept Form: Slope-intercept form is a way of representing a linear equation in two variables, typically written as $y = mx + b$, where $m$ represents the slope of the line and $b$ represents the $y$-intercept. This form allows for easy interpretation of the line's characteristics and is widely used in the study of linear functions and their applications.
Solution: A solution is a homogeneous mixture composed of two or more substances. In a solution, a solute is dissolved in a solvent, resulting in a single phase with a uniform composition and properties throughout.
Solution set: A solution set is the set of all possible values that satisfy a given equation or system of equations. It represents all the solutions that make the equation(s) true.
Solving systems of linear equations: Solving systems of linear equations involves finding the values of variables that satisfy all equations in the system simultaneously. Typically, these systems can be solved using methods such as graphing, substitution, or elimination.
Variable: A variable is a symbol or letter that represents an unknown or changeable value in a mathematical expression, equation, or function. It serves as a placeholder for a value that can vary or be assigned different values within a given context.
Vertical line: A vertical line is a straight line that goes up and down and has an undefined slope. It is represented by the equation $x = a$ where $a$ is a constant.
Y-intercept: The y-intercept is the point at which a line or curve intersects the y-axis, representing the value of the dependent variable (y) when the independent variable (x) is zero. It is a crucial concept in understanding the behavior and properties of various mathematical functions and their graphical representations.
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2.3 Models and Applications