Problem 7 - Algebra 1: Concepts and Skills

Question

Use linear combinations to solve the linear system. Then check your solution. \(x+y=4\) \(x-y=-10\)

Step-by-Step Solution

Answer

The solution to the system of equations is \(x=-3, y=7\).

Step 1: Setup the system of equations

The given system of equations is: \n\(x+y=4\) \n\(x-y=-10\)

Step 2: Combine the equations

We can obtain a linear combination by adding the two equations together: \n\(x+y+x-y=4+(-10)\) \nThis simplifies to \(2x=-6\)

Step 3: Solve for x

Solve the equation for x by dividing both sides by 2: \n\(2x/2 = -6/2\) \nThis gives \(x=-3\)

Step 4: Substitute x into one of the original equations

We can now substitute \(x=-3\) into the original equation \(x+y=4\): \n\(-3+y=4\)

Step 5: Solve for y

By solving for y in the equation \(-3+y=4\), we obtain: \n\(y=7\)

Step 6: Verify the solution

Check that the solution \(x=-3, y=7\) works in both original equations: \nFor \(x+y=4\), substituting x and y we get \(-3+7=4\), which is correct. \nFor \(x-y=-10\), substituting x and y we get \(-3-7=-10\), which is also correct. So the solution \(x=-3, y=7\) is correct.

Key Concepts

Linear Combinations

In the realm of mathematics, linear combinations are a fundamental concept used to solve systems of equations. The term "linear combination" refers to the process of adding two or more equations together, potentially after multiplying them by certain numbers (also known as coefficients). This technique can reveal new equations that help to simplify a system. When dealing with linear equations like:

• \(x + y = 4\)
• \(x - y = -10\)

The main goal is to eliminate one of the variables, making it easier to solve for the other. By adding these two equations, the \(y\) terms cancel each other out, leaving you with a single equation in terms of \(x\): \[x + y + x - y = 4 + (-10)\]Which simplifies to:\[2x = -6\]This simplification is a direct result of using a linear combination. Note that sometimes you may need to multiply one or both equations by particular numbers to line up the coefficients correctly, but in this case, addition worked without further adjustments.

Solving Equations

Once linear combinations have simplified the system of equations, the next step is often more straightforward. You solve for one of the variables by isolating it on one side of the equation. From our simplified equation:\[2x = -6\]We divide both sides by 2 to isolate \(x\):\[x = \frac{-6}{2}\]This gives:\[x = -3\]With \(x\) known, substitute it back into one of the original equations to solve for the other variable. Using \(x + y = 4\), replace \(x\) with \(-3\):\[-3 + y = 4\]Add 3 to both sides to solve for \(y\):\[y = 7\]This steady process shows the beauty of linear systems, where finding one variable affords us the immediate path to the other's value. Each solution step depends on algebra rules, highlighting systematic solving strategies.

Verification of Solutions

After solving the system of equations, an important step follows: verifying the solutions. This ensures that the values discovered indeed satisfy the original equations. In our case, with solutions \(x = -3\) and \(y = 7\), you must check each against both original equations:

• For the equation \(x + y = 4\): Substitute \(x = -3\) and \(y = 7\), thus:\[-3 + 7 = 4\]This statement holds true as both sides equal 4.
• For the equation \(x - y = -10\): Substitute \(x = -3\) and \(y = 7\), resulting in:\[-3 - 7 = -10\]Here too, the statement is correct, confirming our solution.

Verification acts like a safety check to confirm the accuracy of your steps and results. Confirming the solution with all original equations is crucial, ensuring no errors in calculations along the way. This practice fosters confidence and reliability in solving linear systems.