Linear Equations

A linear equation is an algebraic equation where each term is either a constant or the product of a constant and a single variable raised to the first power. The graph of a linear equation is a straight line. The most common form is the slope-intercept form, y=mx+b, where:

• x and y are the variables.

• m is the slope, which represents the steepness of the line.

• b is the y-intercept, the point where the line crosses the y-axis.

Solving Linear Equations

Solving a linear equation means finding the value of the variable that makes the equation true. The goal is to isolate the variable on one side of the equation. This is achieved by performing the same operation (addition, subtraction, multiplication, or division) to both sides of the equation to maintain equality.

Example

Let's solve the equation 2x+5=11 for x.

1. Subtract 5 from both sides: 2x+5−5=11−5 2x=6

2. Divide both sides by 2: 22x​=26​ x=3

The solution is x=3. To check your answer, substitute 3 back into the original equation: 2(3)+5=6+5=11. This confirms the solution is correct.

Systems of Linear Equations

A system of linear equations is a set of two or more linear equations with the same variables. The solution to a system is the point (x,y) that satisfies all the equations simultaneously. Graphically, this is the point where the lines intersect.

There are three possible outcomes when solving a system of two linear equations:

1. One unique solution: The lines intersect at a single point.

2. No solution: The lines are parallel and never intersect.

3. Infinitely many solutions: The lines are identical, meaning they overlap at every point.

Common methods for solving systems of linear equations include:

• Substitution: Solve one equation for one variable and substitute that expression into the other equation.

• Elimination: Multiply one or both equations by a constant so that when the equations are added or subtracted, one of the variables is eliminated.

• Graphing: Graph both lines and find their point of intersection.