Solutions to Linear Equations
Solutions to linear equations are values of the variable that satisfy the equation, making it true. Linear equations are a type of algebraic equation characterized by a constant rate of change, represented graphically as straight lines on a coordinate plane. They can involve one or more variables, and the solutions can vary: some equations may have a single solution, none at all (known as a contradiction), or infinitely many solutions (called identities). For instance, the equation \(3x + 1 = 5\) has a unique solution, \(x = 2\), while the equation \(x + 2 = x + 5\) has no solutions.
In the case of equations with two variables, such as \(y = 2x + 4\), solutions are pairs of numbers that correspond to points on a graph, forming a line when plotted. Key attributes of these graphs include the x-intercept and y-intercept, where the graph intersects the x-axis and y-axis, respectively. The x-intercept can be found by setting \(y\) to zero, while the y-intercept is determined by setting \(x\) to zero. Understanding these concepts is essential for solving linear equations and interpreting their graphical representations.
Subject Terms
Solutions to Linear Equations
A relationship between two quantities can be expressed using a mathematical equation, or a statement that two quantities are equal. Linear equations represent relationships involving a constant rate and have numerous applications. For example, phone bills that charge a constant price per text or megabyte of data, memberships that charge a constant price per month, vehicles traveling at a constant number of miles per hour, and payment schemes that charge a constant number of dollars per unit are all easily modeled using linear equations.
Overview
In an algebraic equation, at least one quantity in the equation varies; this quantity is called a variable. Linear equations are a particular type of algebraic equation in which all permissible values of the variables take the form of a line on the coordinate plane. When written symbolically, the variable(s) in a linear equation is raised to the first power.
A linear equation may have one variable or several variables, depending on the number of dimensions represented by the equation. Solutions to a linear equation are values of the variable that makes the equation true. For example, in the equation 3x + 1 = 5, the number 1 is not a solution because 3(1) + 1 ≠5, but the number 2 is a solution because 3(2) + 1 = 5. Linear equations with one variable may have zero solutions, one solution, or an infinite number of solutions. The equation 3x + 1 = 5 has one solution because the number 2 is the only real number that can be substituted for x such that the equation is true. A linear equation in which no values of the variable result in a true statement, as is the case in the equation x + 2 = x + 5, has no solutions and is called a contradiction. A linear equation that is true for all real values of the variable, such as x + 2 = x + 2 or 2(x + 1) = 2x + 2, has an infinite number of solutions and is called an identity. Solutions to a linear equation in one variable are commonly represented on a number line.
Solutions to linear equations with two variables, like the equation y = 2x + 4, are pairs of corresponding numbers (x,y) that make the equation true. For example, (1,6) is a solution to the equation because 5 = 2(1) + 4, but (1,7) is not a solution because 7 ≠ 2(1) + 4. An infinite number of pairs satisfy linear equations with two variables, so an infinite number of solutions exist. When these pairs of numbers are graphed on a (x,y) coordinate plane, the points form a line. Some characteristics of a graph of a linear equation with two variables have particular significance, such as the point where the line intersects the x-axis (x-intercept) and the point where the line intersects the y-axis (y-intercept).
Solving for the x-Intercept
The x-intercept of a linear equation is the point on a coordinate plane where the graph of the equation intersects the x-axis. The y-coordinate of all points on the x-axis is 0 so the x-intercept of a linear equation can be found by substituting the number 0 for the y-value of the equation. For example, (−1/2,0) is the x-intercept of the linear equation y = 2x + 4 because 0 = 2(−1/2) + 4.
Solving for the y-Intercept
The y-intercept of a linear equation is the point on a coordinate plane where the graph of the equation intersects the y-axis. The x-coordinate of all points on the y-axis is 0 so the y-intercept of a linear equation can be found by substituting the number 0 for the x-value of the equation. For example, (0,4) is the y-intercept of the linear equation y = 2x + 4 because 4 = 2(0) + 4.
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