Linear Equation Definition

Linear Equation Definition

An equation relies on the word “equal” so an equation is simply a math statement that includes two, or more, things being set equal to each other. For example, 3+2 is not an equation, but 3+2=5 is an equation simply based on the equal sign. This holds true even if it uses letters, symbols, or the Greek alphabet instead of numbers.

Something linear is related to a line. So a linear equation is an equation that when graphed makes a line. A linear equation can have one, two, three, or more variables.

Graph of two linear equations
two linear equations, linear equation graph

 

Forms of Linear Equations

The general form of a linear equation is y = mx + c. This equation gives us some key information about the linear equation meaning. The slope is represented by m and the x-intercept is represented by c. So in the equation y = 3x + 2 it is already known that there will be one point at (0, 2) and the slope will be 3. Remember slope is “rise over run” so a slope of 3 would mean rising or going up 3 and running or going to the right 1. From this, another point on this line can be found by adding 3 to the y value (the rise) and 1 to the x value (the run). Another point on this line would be (1, 5). If the dots are connected and the line extended in both directions, a graph of the linear equations has been achieved.

The general form of the linear equation can be re arranged so it might look different. It might appear as:

  • y – c = mx
  • y – mx = c
  • y – mx – c = 0
  • c + mx = y

Each of these can be rearranged to get it back to the more desirable y = mx + c format. Other linear equations might appear to be missing a part: y = c or y = mx, for example. In these cases, the missing part must be equal to 0. If the linear function is in y = c format or even x = any number this is considered a constant function or identity function. If the y = c then the line will be horizontal or flat. For every value of x the y value will be the same. If it is x = any number then the line will be vertical. For any value of y the x value will remain the same.

Linear Equation Constant Function
Linear Equation Constant Function

 

Linear Equation Examples

The easiest linear equations are the constant function or the identity function where y = c or x = any number. In real life, this might be an age in years for any month that doesn’t include that person’s birthday. It will remain constant throughout the month. Here are some linear equation examples:

  • y = 15
  • x = 32

Two variable linear equations fit the general form of y = mx + c. In real life this might help someone determine the cost of a ride share. For example, if it costs $1 for the driver to pick up plus $0.20 per mile, a linear equation of y = .2x + 1 can be created to determine the cost. Here are some other linear equation examples with two variables:

  • y = 1 + .2x
  • .2x = y – 1
  • .2x – y + 1 = 0

Just because an equation has two variables doesn’t always mean it will be linear. There could also be a parabola (y = x squared – 3) or a circle ( y = 2 = the square root of (25 -x)). Remember if the graph has any type of curve and not just a line, then it will not be considered linear. We call these types of equations non-linear.

Three variable linear equations in real life might include trying to meal plan when limited on calories, fat, and sugar. For example, if only three meals are eaten with no snacks, and the limit of calories is 2,000, the limit of fat is 20 grams, and the limit of sugar is 20 grams, a system of linear equations can be created. Assume b stands for breakfast, l for lunch, and d for dinner.

  • b + l + d = 2000 calories
  • b + l + d = 20 grams of fat
  • b + l + d = 20 grams of sugar

For linear equation examples with four or more variables, it is apparent that someone could simply add more meals, snacks, or drinks to the above equations. They could also subtract calories burned off when working out.

  • b + l + d + s = 2000 calories
  • b + l + d + s – exercise = 2000 calories

System of Linear Equations

system of linear equations is a group of linear equations that work together. These can be graphed to solve for different information. Let’s look at some system of linear equation examples.

I want to use rideshare and I have two different apps available. One has a higher initial cost to pick me up than the other, but it also has a lower cost per mile. I need to know which one will cost me the least amount of money. Option A: Pick up costs $1 and $0.20 per mile. Option B: Pick up costs $2 and $0.15 per mile. I can use this information to create two linear equations.

  • y = .2x + 1
  • y = .15x + 2

I could plug in the number of miles for each to determine which is cheaper this time. Or I could solve the system of equations so I will always know at what point I should switch from one company to the other.

I want to open a lemonade stand and I need to buy lemons, sugar, and cups. I have $20 to spend. What is the maximum amount I can buy?

  • l + s + c = $20

For every 6 servings, I need 6 lemons, 1 cup of sugar, and 6 cups. I need to balance my shopping to maximize the amount of lemonade I can make. I don’t want to end up with extra lemons and sugar and no cups to serve it in. These are just some examples of when it is helpful to understand systems of linear equations.

Solution of Linear Systems

If we are told all equations are said to be true, then there are three possible outcomes when solving a system of linear equations. It might be found that there are no solutions, exactly one solution, or there is an infinite number of solutions. This is true, no matter which method is used to solve the system of equations:

  • Graphing
  • Substitution
  • Elimination
  • Row Reduction
  • Determinant

If all of the equations given can be rearranged into the exact same equation or graphed and displayed as the same line, then one can say there are infinitely many solutions. It doesn’t matter what number is put into it, a solution to the linear system will always be able to be found. This is considered a dependent system of equations.

Infinite Solutions to a Linear Equation
Linear Equation All Solutions Infinite
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