# Systems of linear equations

• elizavikki
In summary, systems of linear equations are mathematical equations that involve two or more linear equations with the same variables. They are used to represent relationships between variables and can be solved to find the values of the variables that satisfy all the equations. A linear equation describes a straight line on a graph with one variable, while a system of linear equations involves multiple equations with the same variables. The solution to a system of linear equations is the point of intersection between the lines represented by the equations and can be found using methods such as substitution, elimination, and graphing. Real-life applications of systems of linear equations include economics, engineering, physics, and data analysis.

#### elizavikki

Can anyone help me to learn how to solve systems of linear equations?
{-5x-5y=-5
{-40x-3y=2

$$a+b=c$$
$$2a+b=3c$$

Eliminate either a or b. Since b is easier to get rid of, multiply either the 1st or 2nd equation by a negative. Then simply add straight down as you would normally.

Where is your work? I'm not going to work your problem, just giving you an example. So figure it out from there.

Sure, I'd be happy to help you learn how to solve systems of linear equations using the addition method. First, let's review what a system of linear equations is. It is a set of two or more equations with two or more variables, where the solution is the set of values that makes all of the equations true at the same time. The addition method, also known as the elimination method, involves adding or subtracting the equations in the system to eliminate one of the variables, ultimately allowing you to solve for the remaining variable. To apply this method to the system you provided, we can start by multiplying the first equation by 8 and the second equation by 5 to create opposites for the x variable. This will result in the following equations:

-40x - 40y = -40
-40x - 15y = 10

Next, we can add these equations together, which will eliminate the x variable:

-40x - 40y = -40
+ (-40x - 15y = 10)
-----------------------------
-55y = -30

Now, we can solve for y by dividing both sides by -55:

y = (-30)/(-55) = 6/11

To find the value of x, we can substitute this value for y into one of the original equations. Let's use the first equation:

-5x - 5(6/11) = -5
-5x = -5 + 30/11
-5x = -55/11 + 30/11
-5x = -25/11
x = (-25/11)/(-5) = 5/11

So, the solution to this system of equations is x = 5/11 and y = 6/11. I hope this helps you understand how to use the addition method to solve systems of linear equations. Keep practicing and you will become more comfortable with this method!

## What are systems of linear equations?

Systems of linear equations are mathematical equations that involve two or more linear equations with the same variables. These equations are used to represent a relationship between different variables and can be solved to find the values of the variables that satisfy all the equations.

## What is the difference between a linear equation and a system of linear equations?

A linear equation is a mathematical expression that describes a straight line on a graph. It has only one variable and can be solved to find the value of that variable. A system of linear equations, on the other hand, involves two or more linear equations with the same variables and can be solved to find the values of all the variables that satisfy all the equations.

## What is the solution to a system of linear equations?

The solution to a system of linear equations is the set of values for the variables that make all the equations in the system true. In other words, it is the point or points of intersection between the lines represented by the equations. A system of linear equations can have one unique solution, no solution, or infinitely many solutions.

## How can a system of linear equations be solved?

There are several methods for solving a system of linear equations, including substitution, elimination, and graphing. In substitution, one equation is solved for one variable and then substituted into the other equation. In elimination, the equations are manipulated to eliminate one of the variables. In graphing, the equations are plotted on a graph and the point of intersection is found. Other methods, such as matrices and determinants, can also be used.

## What are some real-life applications of systems of linear equations?

Systems of linear equations have many real-life applications, such as in economics, engineering, and physics. They can be used to model and solve problems involving multiple variables, such as budgeting, production planning, and electrical circuits. They are also used in data analysis and prediction, such as in regression analysis and linear programming.