Question

Use linear combinations to solve the system of linear equations. $$\begin{aligned} &2 x+y=4\\\ &x-y=2 \end{aligned}$$

Short answer

The solution to the system is \(x = 2\), \(y = 0\)

Step-by-step solution

Normalize equations

The equations are already in standard form: \[2x + y = 4\] and \[x - y = 2\]. This places the problem in a good position to begin with.

Add equations

Add the two equations to eliminate \(y\). Adding the two gives: \[3x = 6\]. This simplifies to: \[x = 2\]. This works as a linear combination here effectively cancels out \(y\)

Substitute \(x = 2\) into second equation

Substitute \(x = 2\) into the second equation: \(2 - y = 2\), which simplifies to: \(y = 0\)

Verification

Substitute \(x = 2\) and \(y = 0\) back into the original equations to verify the solution: Both \[2(2) + 0 = 4\] and \[2 - 0 = 2\] are correct, so the solution is valid.

Key concepts

Linear Combinations

Linear combinations represent a cornerstone in understanding systems of linear equations. At its core, the method involves combining equations in a way that allows you to isolate one variable. In the case of our exercise, the equations are given as \[2x + y = 4\] and \[x - y = 2\]. To employ linear combinations, you look for coefficients of variables that can be added or subtracted to eliminate a variable. Here, simply adding the given equations results in the variable \(y\) canceling out, leaving you with an equation in \(x\) only: \[3x = 6\]. You solve for \(x\) and use this value to then determine \(y\).

Advantages of Linear Combinations

Using linear combinations can be a quick way to solve a system, especially when the system is already set up nicely as in our exercise. It's like puzzle solving, where you look for the pieces (equations) that fit together to simplify the problem. This method is particularly useful when dealing with larger systems of equations, as it can often save time and calculations.

Substitution Method

When linear combinations might seem too complex or when you're faced with equations that don't lend themselves well to that method, the substitution method is another tool at your disposal. It's all about 'substituting' one variable with an equivalent expression from another equation.

In the context of our exercise, after finding \(x = 2\) from the addition of both equations, you substitute this value into the second equation: \[x - y = 2\] to get \[2 - y = 2\], which simplifies to \[y = 0\]. Substituting eliminates variables sequentially, systematically reducing the system to equations in fewer variables.

Why Use Substitution?

It can be incredibly helpful when equations are not structured for easy combination, or if you're dealing with non-linear systems. Substitution gives you a direct method to find one variable at a time, often leading to a clear path to the solution.

Verification of Solutions

A solution isn't complete without verification. It's the process of ensuring that the answers you have found indeed satisfy all of the original equations. After solving for \(x\) and \(y\) by either linear combinations or substitution, we take those values and plug them back into the original equations.

In our example, substituting \(x = 2\) and \(y = 0\) back into the given equations gives: \[2(2) + 0 = 4\] and \[2 - 0 = 2\], which verifies that the solution is correct. If we didn't verify, we could potentially overlook an incorrect assumption or calculation error that would render our solution moot.

Always Verify!

The verification step is crucial in mathematics. It not only confirms the correctness but also enhances your understanding of the relationships between equations and their solutions. It is the seal of approval on your hard work.