Problem 21 - Elementary Algebra Within Reach

Question

In Exercises 21-26, solve the system by the method of substitution. $$ \left\\{\begin{array}{l} 4 x-y=2 \\ 2 x-\frac{1}{2} y=1 \end{array}\right. $$

Step-by-Step Solution

Answer

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

Step 1: Isolate a Variable in one equation

Firstly, isolate one of the variables, say \( y \), in the first equation. Substract \( 4x \) from both sides of Equation 1, which gives: \( y = 4x - 2 \)

Step 2: Substitute \( y \) in the second equation

Now substitute \( y \) from the first equation into the second equation: \( 2x - \frac{1}{2} (4x-2) = 1 \). Simplify the equation, multiplying through the brackets and simplifying gives: \( 2x - 2x + 1 = 1 \)

Step 3: Solve for \( x \)

Solve the resulting equation for \( x \). After subtracting \( 2x \) from each side and simplifying, we get: \( x = 0 \)

Step 4: Substitute \( x \) value into the first equation

Now substitute the found value for \( x \) back into the first equation in order to solve for \( y \): \( 4*0 - y = 2 \). Simplify to get the value for \( y \): \( -y=2 \) or \( y=-2 \)

Step 5: Verification

Lastly, check the result by substituting \( y = -2 \) and \( x = 0 \) into both initial equations. Verification shows that the solutions satisfy both equations, hence they are correct

Key Concepts

System of Equations

A system of equations consists of two or more equations with a common set of variables. In our problem, the system includes these two equations:

• \(4x - y = 2\)
• \(2x - \frac{1}{2}y = 1\)

The goal is to find values for \(x\) and \(y\) that satisfy both equations simultaneously. This method offers insights by illustrating how variables interact in a unified context. By treating the equations as a system, we can solve complex problems in a structured manner. It's an essential skill in algebra, applied extensively in science and engineering.

Algebraic Solutions

An algebraic solution involves manipulating equations using algebraic operations to find the values of unknowns. The method of substitution is particularly useful for solving systems of equations.Firstly, we isolate one variable in one equation. In the given exercise, we isolate \(y\) in the first equation by rearranging it to get \(y = 4x - 2\). Next, we substitute this expression for \(y\) into the second equation. With substitution, the second equation becomes \(2x - \frac{1}{2}(4x -2) = 1\). This way, we convert the second equation to only include one variable \(x\), allowing us to solve it algebraically to find \(x = 0\). Solving equations by substitution simplifies complex problems into manageable steps.

Verification of Solutions

Verification of solutions is an essential step to confirm the correctness of the solution derived from the equations. After finding \(x = 0\) and \(y = -2\), it's important to check if these values satisfy both original equations.Substitute \(x = 0\) and \(y = -2\) back into:

• \(4x - y = 2\): Here, \(4(0) - (-2) = 0 + 2 = 2\), which matches the original equation.
• \(2x - \frac{1}{2}y = 1\): Thus, \(2(0) - \frac{1}{2}(-2) = 0 + 1 = 1\), which also holds true.

Both equations are satisfied, verifying our solution. This step is crucial in ensuring the solution's accuracy before concluding the problem-solving process.

Elementary Algebra

Elementary algebra forms the foundation for solving equations and inequalities, focusing on fundamental operations here like addition, subtraction, multiplication, and division.Understanding how to manipulate and rearrange equations is a core part of elementary algebra. In this exercise, after isolating \(y\), we substituted it into the other equation. Each step requires an understanding of how algebraic operations interact to simplify expressions.Key techniques like distributing numbers over parentheses and combining like terms are used here. Elementary algebra helps students build a toolkit for approaching and solving more complex equations in higher mathematics.