Solving Linear Equations
Mini Lecture Video
Key Takeaways
A Linear Equation is an equation involving monomials of degree no greater than \(1\). In other words, a linear equation is nothing more than an equation involving multiples of \(x\) and real numbers.
Ex: \(x+3=5x-\frac{3}{2}\).
Solving linear equations requires nothing more than getting \(x\) by itself on one side of the equation so that there is a number on the other side of the equation using algebraic operations.
In general, when solving equations, the goal is to find specific values for the variables in the equation so that the equation is true. Looking at the example \(x+3=5x-\frac{3}{2}\), this equation is not always true for every \(x\) you plug into both sides (let \(x=1\) on both sides and you will see this is correct). However, there are specific \(x\)-values for which the left-hand side (LHS) is equal to the right-hand side (RHS). Those specific values are called an equation’s solution set. Getting \(x\) by itself on one side of the equation will give you the solution set for the equation.
Note that not every equation must yield a solution for \(x\). Some equations have no solution. For example, \(3x+2=3x+1\) has no solution, because when you subtract \(3x\) from both sides, you will have \(2=1\) which is a contradiction, (i.e. a mathematical statement that is false). Every equation that you attempt to solve that has no solution will yield a contradiction.
Some equations have many solutions… sometimes infinitely many. For example, \(2x+1=\frac{1}{2}x+\frac{3}{2}x+1\) is a true equation no matter what \(x\) is. “Why,” you ask? Combine like-terms on the RHS, then the LHS and RHS are the same. Since \(x\) represents an arbitrary unknown number and the RHS=LHS, it doesn’t matter what \(x\) you plug in, the equation will be true regardless.