Problem 9 - Algebra 2

Question

Solve each system of equations by using elimination. \(\frac{1}{4} x+y=\frac{11}{4}\) \(x-\frac{1}{2} y=2\)

Step-by-Step Solution

Answer

\(x = 3\), \(y = 2\).

Step 1: Clear Fractions

To eliminate fractions, multiply the entire first equation by 4 to get rid of the fraction on the variable and the constant. Doing this gives us:\( x + 4y = 11 \)

Step 2: Adjust the Second Equation

To facilitate the elimination process, let's make the second variable consistent with the new 'y' term in our first equation by multiplying the second equation by 2. Doing this gives us:\( 2x - y = 4 \)

Step 3: Eliminate y

Add the two equations together such that the \(y\) terms cancel each other: \( x + 4y = 11 \)\( 2x - y = 4 \) Adding these gives: \( 3x + 3y = 15 \)

Step 4: Solve for x

Simplify the result from the addition to solve for \(x\):\( 3x + 3y = 15 \) This can be divided entirely by 3 giving: \( x + y = 5 \)

Step 5: Substitute and Solve for y

Substitute \(x = 5 - y\) into one of the initial equations, such as the modified second equation:\( 2(5-y) - y = 4 \)Simplify and solve for \(y\):\( 10 - 2y - y = 4 \)\( 10 - 3y = 4 \)\( -3y = -6 \)\( y = 2 \)

Step 6: Solve for x Using Solved Value of y

Substitute \(y = 2\) back into \(x = 5 - y\):\( x = 5 - 2 = 3 \)

Step 7: Verify the Solution

Check that \(x = 3\) and \(y = 2\) satisfy both original equations:1. \(\frac{1}{4}(3) + (2) = \frac{3}{4} + 2 = \frac{11}{4}\)2. \((3) - \frac{1}{2}(2) = 3 - 1 = 2\)Both equations are satisfied, so the solution is correct.

Key Concepts

Systems of Equations

Systems of equations are a set of two or more equations that share the same variables. In our context, we are working with linear equations, which means each equation is a line on a graph. The main goal when solving a system of equations is to find values of the variables that satisfy every equation in the system.

• If we have two equations and plot them on a graph, the solution represents the point where these lines intersect. This intersection point gives us the values of the variables, such as \( (x, y) \).
• Understanding these solutions graphically is important. It helps to visualize why certain methods like elimination work.

There are various methods to solve systems of equations, and one popular method is the elimination method. This approach focuses on eliminating one variable by combining the equations. This method results in a single equation that you can solve directly. Understanding systems of equations lays the groundwork for many real-world problems, from figuring out supply and demand in economics to optimization problems in engineering.

Solving Linear Equations

Solving linear equations involves finding the values of the variables that make the equation true. In a system of equations, we often have more than one equation to consider at a time. The elimination method is particularly useful because:

• It allows for simplifying the system of equations step-by-step.
• We can strategically eliminate one variable, reducing the complexity of the problem.

Let's think about this through our example: When tackling the problem, the first step was to remove the fractions by multiplying through each term in the equation. This step makes equations easier to handle. It's beneficial because it turns fractions into integer coefficients, simplifying algebraic manipulation in subsequent steps. Remember, solving these equations step-by-step helps in catching and correcting mistakes early in the process, which is crucial for success when you deal with more complex systems or more than two variables.

Algebraic Manipulation

Algebraic manipulation is the toolkit you use to transform equations into solvable forms. In the elimination method, we applied various algebraic manipulations:

• We first cleared fractions to get a clean, easier-to-manage equation. This step is crucial as fractions introduce complications that can lead to mistakes later.
• We adjusted coefficients to help eliminate variables by multiplication, creating terms with equal and opposite signs.
• Finally, we combined the adjusted equations, which allowed us to eliminate one of the variables.

These steps highlight the core principle of algebraic manipulation: "Simplify the problem while retaining the solution." Every manipulation should bring us closer to isolating the desired variable or making mathematical operations more straightforward. Algebraic manipulation in this context can involve addition, subtraction, multiplication, or division. Remember that each manipulation needs to maintain the equation's balance, which means whatever you do to one side, you must do to the other. Mastering these basics develops a strong foundation for tackling more complex algebraic equations in future studies.