Math 103/Chapter 8.1

What is a linear equation in two variables?

A linear equation in two variables describes a relationship in which the value of one of the variables depends on the value of the other variable.

What is a linear equation?

A linear equation looks like any other equation. It is made up of two expressions set equal to each other. A linear equation is special because: It has one or two variables. No variable in a linear equation is raised to a power greater than 1 or used as the denominator of a fraction. When you find pairs of values that make the linear equation true and plot those pairs on a coordinate grid, all of the points for any one equation lie on the same line. Linear equations graph as straight lines.

A word of caution...

A word of caution in an equation in which an absolute value is equal to a negative number has no solution. Do NOT try to rewrite the equation as two equivalent equations.

What is an absolute value equation?

An absolute value equation is a term used in mathematics to indicate the distance from a point or number from the origin or "zero point" of a number line.

Solve I2x-3I=7

Solution begin by converting the absolute value to TWO equations the do NOT involve absolute values. I2x-3I=7 1.) 2x-3=7 OR 2x-3=-7 Convert to TWO linear equations. 2.) Add "3" to both sides of each equation. 2x=10 or 2x=-4 3.) Divide both sides of each equation by 2. x=5 or x=-2.

Solve IxI=5

Solution recall the the absolute value of a number x denoted IxI, is a measure of the distance between 0 and that number x on a real number line. In this equation, we are looking for a number x that is 5 units from 0. There are two such numbers {-5,5}.

Solve 2x-6=10

Solve 2x-6=10 1.) begin by isolating the variable term on the left side of the equation. 2x-6=10 1.) add "6" to both sides to isolate 2x. 2x=16 2.) divide both sides by 2 to get "X" alone. x=8 the solution to the set is {8}.

Solve 3(2x+7)-4x=5x-6

Solve 3(2x+7)-4x=5x-6 1.) Distribute the "3" to remove the parantheses. 6x+21-4x=5x-6. 2.) Combine like terms. 2x+21=5x-6 3.) Collect variable terms on the right side by subtracting "2x" from both sides. Collect constant on the left side by adding "6" to both sides. 27=3x. 4.) Divide both sides by "3" 9=x. The solution to the set is {9}.

Solve 4/3x+5/6=9/4x-1

Solve 4/3x+5/6x=9/4x-1. Both sides contain fractions, we can clear these fractions by multiplying both sides by the LCM of 12. 12(4/3x+5/6) =12(9/4x-1) 2.) Distribute and divide out common factors. 12/1(4/3x)+12/1(5/6) =12/1(9/4x)-12(1). 3.) Multiply 16x+10=27x-12. 4.) Subtract 16x from both sides. 5.) Add "12" to both sides 22=11x. 6.) Divide both sides by "11". 2=x. The solution to the set is {2}.

Solving absolute value equations

Solving absolute value equations: For any expression X and any positive number a, the solutions to the equation. IXI=a can be found by solving the two equations X=a and X=-a.

What are the steps to solving a linear equation?

Steps in solving a linear equation: 1.) Simplify each side of the equation. This includes performing any distributive multiplications, clearing all fractions by multiplying each side by the (LCM) least common multiple of the denominators and combining like terms. 2.) Collect all variable terms on one side of the equation and all constant terms on the other side. It is a good idea to collect the variable terms in such a way that the coefficient of the variable term will be positive. 3.) Divide both sides of the equation by the coefficient of the variable term to find the solution. 4.) Check your solution and write it in a solution set notation.