Problem 39 - Algebra 1: Concepts and Skills

Question

Use linear combinations to solve the linear system. Then check your solution. \(4 x+3 y=1\) \(2 x-3 y=1\)

Step-by-Step Solution

Answer

The solution for the system of equations is \(x = 1/3\) and \(y = -1/9\)

Step 1: Line up the equations

Start by arranging the equations so that like terms are vertically aligned:\n \(4x + 3y = 1\) \n \(2x - 3y = 1\)

Step 2: Combine the equations

After arranging the equations, add them together. Target to add in such a way that the y terms will cancel out. The sum of the equations is: \n \(4x + 2x = 1 + 1\) \n which simplifies to \n \(6x = 2\)

Step 3: Solve for x

Solve for x by dividing both sides by 6, to get x isolated: \n \(x = 2/6 = 1/3\)

Step 4: Substitute x into the original equation

Now, substitute the value of x into one of the original equation. Let's take the first one: \n \(4*(1/3) + 3y = 1\)

Step 5: Solve for y

Simplify and solve the above equation for y: \n \(4/3 + 3y = 1\) \n subtract \(4/3\) from both sides leads to \n \(3y = 1 - 4/3 = -1/3\) \n Divide both sides by 3 to obtain: \n \(y = -1/9\)

Step 6: Check the solution

The last step is to confirm if the found x and y values satisfy both of the original equations. So, substitute \(x = 1/3\) and \(y = -1/9 \) into both of the original equations and check if both are true. If yes, that means the solution is correct.

Key Concepts

Linear Combinations

In the context of linear equations, a linear combination involves adding or subtracting equations in a way that eliminates one of the variables. This strategy simplifies the problem, allowing us to solve for the remaining variable. Consider our system of equations:

• \(4x + 3y = 1\)
• \(2x - 3y = 1\)

By adding these two equations, notice the \(y\) terms cancel each other out. This clever manipulation leaves us with \(6x = 2\). Why is this step crucial? It transforms our two-variable problem into a single variable equation. Thus, we can solve for \(x\) without interference from \(y\). Linear combinations are a powerful technique, offering a shortcut in multi-variable equations.

Solving Systems

Solving a system of linear equations means identifying the values of the variables that satisfy all equations involved. Once a linear combination has been used, our system simplifies, as shown by the equation \(6x = 2\). Dividing each side by 6 gives \(x = \frac{1}{3}\). This value must be checked in both original equations, but first, we complete the system by solving for \(y\). Substitute \(x = \frac{1}{3}\) back into one of the original equations, say \(4x + 3y = 1\). This substitution leads to \(4(\frac{1}{3}) + 3y = 1\), simplifying to \(3y = -\frac{1}{3}\). Finally, dividing yields \(y = -\frac{1}{9}\). By accurately substituting and simplifying, we solve the system efficiently.

Checking Solutions

Checking your solutions is a key step in solving any system of equations. After finding \(x = \frac{1}{3}\) and \(y = -\frac{1}{9}\), substitute these values back into both original equations to ensure they hold true. For the first equation:

• \(4(\frac{1}{3}) + 3(-\frac{1}{9}) = 1\)

Simplify to verify the left side equals 1. Similarly, check the second equation:

• \(2(\frac{1}{3}) - 3(-\frac{1}{9}) = 1\)

Successful verification confirms the solution is correct for both equations.Why is double-checking so important? It ensures that the values of \(x\) and \(y\) truly solve each equation, guaranteeing the accuracy of your results. Always take this final step to confirm your solutions.