Introduction to Linear Combination Calculator With Steps:
Linear Combination Calculator is an online tool that helps you to find the system of linear equations solution in less than a minute. It computes two linear equations to evaluate the unknown variable values.
This linear combinations calculator is a great source for solving linear equations without doing any manual calculations on a platform without going to any teacher.
What is a Linear Combination?
Linear combination is a process that is used in algebra to evaluate the value of unknown variables that are present in linear system equations. You can use this method to find one, two, or even three unknown variable values.
To get the solution of variables x, y, and z from the linear equation you can use the substitution method, elimination method, cross multiplication method, and graphing linear equation method.
Linear Equations are:
The linear equation is the equation that has order and degree one, and you can add different numbers of variables. Only two variables are involved in the given equation: x and y. Whereas a, b, d, and e are coefficients of x and y, and c and f are constants,
$$ ax + by \;=\; c $$
$$ dx + ey \;=\; f $$
Working of Linear Combination of Matrices Calculator:
Linear combination of matrix calculator use a simple and easy way to solve linear systems of equations involving finding the values of variables that satisfy the given equations simultaneously.
There are primarily two common methods to solve such systems: substitution and elimination. Here's how each method works:
Substitution Method:
To understand how the substitution method works to simplify the linear equation, suppose an example.
$$ \biggr[ \begin{matrix} 2x + 3y \;=\; 10 \\ x - y \;=\; 1 \\ \end{matrix} $$
Step 1: First, choose one of the equations and solve it for one variable. This means isolating one variable on one side of the equation to find the other variable value. Let's choose equation 2 because it is simplest than the first equation x−y=1, solve for x
Step 2: Separate the term x from the above equation that you choose as
$$ x - y \;=\; 1,\; x \;=\; 1 + y $$
Step 3: Now substitute x=y+1 into the first equation 2x+3y=10 and solve it to get the value of variable y.
$$ 2(y + 1) + 3y \;=\; 10 $$
$$ 2y + 2 + 3y \;=\; 10 $$
$$ 5y + 2 \;=\; 10 $$
$$ 5y \;=\; 10 - 2 $$
Divided by 5 on both sides to find the value of y which is y=8/5.
$$ 5y \;=\; 8 $$
Step 4: Substitute y=8/5 into the above equation to get the values of the x variable.
$$ x \;=\; y + 1 $$
$$ x \;=\; \frac{8}{5} + 1 $$
$$ x \;=\; \frac{13}{5} $$
Step 5: Substitute the values of x and y into one of the original equations to ensure they satisfy both equations simultaneously.
$$ 2x + 3y \;=\; 10 $$
$$ 2(\frac{13}{5})+3(\frac{8}{5}) \;=\; 10 $$
$$ \frac{26}{5} + \frac{24}{5} \;=\; 10 $$
$$ \frac{50}{5} \;=\; 10 $$
$$ 10 \;=\; 10 $$
Hence it satisfies the given equation so our values of x and y are true.
Elimination Method:
Let's suppose two linear equations to see how the elimination method work in the evaluation of variable x and y values.
$$ \biggr[ \begin{matrix} 2x & + & 3y & = & 10 \\ x & - & y & = & 1 \\ \end{matrix} $$
Step 1: Arrange the equations such that the coefficients of one of the variables in both equations are equal or opposite. This is achieved by multiplying one or both equations by suitable constants.
Example:
$$ 2x + 3y \;=\; 10,\; x - y \;=\; 1 $$
Multiply the second equation by 2 to make the x variable in both equations.
$$ 2(x - y \;=\; 1) $$
$$ 2x - 2y \;=\; 2 $$
Step 2: Subtract both equations to eliminate one of the variables.
$$ \begin{matrix} 2x & + & 3y & = & 10 \\ \pm 2x & \mp & 2y & = & \pm 2 \\ \hline \\ 0x & + & 5y & = & 8 \\ \end{matrix} $$
$$ y \;=\; \frac{8}{5} $$
Step 3: To solve the equation for the remaining variable, substitute y value in one of the original values.
$$ 2x + 3y \;=\; 10\; put y \;=\; \frac{8}{5} $$
$$ 2x + 3(\frac{8}{5}) \;=\; 10 $$
$$ 2x + \frac{24}{5} \;=\; 10 $$
$$ 2x \;=\; 10 - \frac{24}{5} $$
$$ 2x \;=\; 50 - \frac{24}{5} $$
$$ x \;=\; \frac{26}{5 \times 2} $$
$$ x \;=\; \frac{13}{5} $$
Step 4: To verify the solution, put the values x and y in one of the original equations. If it satisfies then your values are correct.
$$ x - y \;=\; 1,\; x \;=\; \frac{13}{5},\; y \;=\; \frac{8}{5} $$
$$ (\frac{13}{5}) - (\frac{8}{5}) \;=\; 1 $$
$$ \frac{13}{5} - \frac{8}{5} \;=\; 1 $$
$$ 13 - \frac{8}{5} \;=\; 1 $$
$$ \frac{5}{5} \;=\; 1 $$
$$ 1 \;=\; 1 $$
Both methods are effective for solving systems of linear equations, depending on your preference or the specific structure of the equations.
How to Use the Linear Combo Calculator?
Linear combination method calculator has an easy-to-use interface, so you can easily use it to evaluate the linear combination of the given equation solution.
Before adding the input in linear combination calculator matrix for the solutions of given equation problems, you must follow some simple steps. These steps are:
- Add the value of a variable from the first linear system of the equation.
- Add the value of a variable from the second linear system of the equation.
- Recheck your input value for the linear equation solution before hitting the calculate button to start the calculation process in the linear combinations calculator.
- Click on the “Calculate” button to get the desired result of your given linear combination problem.
- If you want to try out our linear combination of matrices calculator to check its accuracy in solution, use the load example.
- Click on the “Recalculate” button to get a new page for solving more linear combination questions.
Final Result of Linear Combination of Matrix Calculator:
Linear combo calculator gives you the solution to a given linear equation problem when you add the input value in it. It provides you solutions . It may contain as:
- Result option
You can click on the result option and it provides you with a solution of linear equation questions.
- Possible step
When you click on the possible steps option it provides you with the solution of the linear equation problem where all calculation steps are included.
Benefits of Linear Combination Method Calculator:
Linear combination calculator gives you multiple benefits whenever you use it to calculate the linear combination equation problems and to get their solutions. These benefits are:
- Our linear combinations calculator saves the time and effort that you consume in solving linear combination questions and get solutions in a few seconds.
- It is a free-of-cost tool that provides you with a solution of a given linear equation to find the value of an unknown variable without paying a single penny.
- It is an adaptive tool that allows you to find the unknown variable (x,y,z) from the given system of linear equations in a linear combination of matrices calculator.
- You can use this linear combination of matrix calculator for practice to get familiar with this concept easily.
- It is a trustworthy tool that provides you with accurate solutions as per your input to calculate the Linear combination problem.
