Linear equations#

are equations with one unknown variable with the power of one and constants.

What is an equation?#

An equation is a mathematical statement that shows that two expressions are equal. It consists of two sides, the left-hand side (LHS) and the right-hand side (RHS), separated by an equal sign (=). The key idea of an equation is to express that the quantities on both sides have the same value.

Equations are used to represent various mathematical relationships and to solve for unknown values. They are commonly used in algebra, physics, engineering, and other scientific disciplines to describe real-world phenomena and to find solutions to problems.

An equation can involve various mathematical operations, such as addition, subtraction, multiplication, division, exponentiation, and more. Solving an equation typically involves finding the values of the variables that make the equation true.

For example, the equation \(8+3x-3=-5x+1\) states that the left-hand side \(8+3x-3\) is equal to the right-hand side \(-5x+1\). Solving this equation involves finding the value of \(x\) that satisfies the equality, which in this case is \(x=-\frac{1}{2}\).

In summary, an equation is a mathematical expression that asserts equality between two sides.

What does it mean to solve an equation?#

Solving equations involves determining the values of variables that satisfy the equality. In other words, find out what is the value of the unknown variable in order to have the equation to be true.

How to solve linear equations?#

  • Perform the same operation to both sides of the equation to maintain equality.

  • Keep track of the properties of the mathematical operations you’re using.

  • Check your solution(s) by substituting them back into the original equation.

Let’s examine the equation mentioned above, \(8+3x-3=-5x+1\). The steps to reach the solution are, for example

\(\begin{align}8+3x-3&=-5x+1 \ \| \ \text{simplify LHS} \\ 3x+5&=-5x+1 \ \|-5 \ \text{Constants can be added to or subtracted from both sides.} \\ 3x+5-5&=-5x+1-5 \ \| \ \text{simplify both sides} \\ 3x&=-5x-4 \ \| +5x \ \text{terms can be added to or subtracted from both sides} \\ 3x+5x&=-5x+5x-4 \ \| \ \text{simplify both sides} \\ 8x&=-4 \ \| \ \text{last step: divide the equations with coefficient of x} \\ \frac{8x}{8}&=\frac{-4}{8} \ \| \ \text{simplify both sides} \\ x&=-\frac{1}{2}\end{align}\)

Check your solution by subsituting the \(x\) in the original equation with your solution. \(\begin{align}8+3x-3&=-5x+1 \ \|x=-\frac{1}{2} \\ 8+3\cdot(-\frac{1}{2})-3&=-5\cdot(-\frac{1}{2})+1 \\ 8-\frac{3}{2}-3&=\frac{5}{2}+1 \\ 5-\frac{3}{2}&=\frac{7}{2} \\ \frac{10}{2}-\frac{3}{2}&=\frac{7}{2} \\ \frac{7}{2}&=\frac{7}{2} \ \text{true}\end{align}\)

Therefore the solution is correct.

Shorter way of solving linear equations#

Transfer terms from one side to the other. As a result, the sign of the term will be reversed. Then simplify.

\(\begin{align}8+3x-3&=6-5x+1 \\ 3x+5&=-5x+1 \\ 3x&=-5x+1-5 \\ 3x&=-5x-4 \\ 3x+5x&=-4 \\ 8x&=-4 \|:8 \\ \frac{8x}{8}&=\frac{-4}{8} \\ x&=-\frac{1}{2}\end{align}\)

Steps are not compulsory, so I (the author) would write down only the following steps:

\(\begin{align}8+3x-3&=-5x+1 \\ 8x&=-4 \|:8 \\ x&=-\frac{4}{8} \\ x&=-\frac{1}{2}\end{align}\)

Examples#

a) \(6x-3=4x+9\)

b) \(\frac{1}{2}x=4\)

c) \(\frac{x}{2}=\frac{2x}{3}+1\)

d) \(5=\frac{7}{x}-2\)

Number of solutions#

There are three possibilities for the number of solutions.

  1. Solution can be an exact one value.

  2. It can be any value (the equation is always true no matter the value of x).

  3. There is no solution at all.

Examples#

We have examined the first case. The second case, when the equation is always true, is for example

\(\begin{align}6x-2&=2x-2+4x \\ 6x-2&=6x-2 \\ 6x-6x&=-2+2 \\ 0&=0 \ \| \ \text{true}\end{align}\)

LHS = RHS no matter the value of variable \(x\).

Answer: Solution exists always and is not dependent on the value of variable \(x\).

The third case, when there is no solution at all, is for example

\(\begin{align}\text{3.)} \ 6x-2&=2x-3+4x \\ 6x-2&=6x-3 \\ 6x-6x&=-3+2 \\ 0&=-1 \ \| \ \text{untrue}\end{align}\)

There exists no value for variable \(x\) so that the equation is true.

Answer: Solution does not exist.