Problem 14 - Thinking Mathematically

Question

Solve each system by the substitution method. Be sure to check all proposed solutions. \(\left\\{\begin{array}{l}x+y=6 \\ y=2 x\end{array}\right.\)

Step-by-Step Solution

Answer

The solution to the system of equations is x = 2, y = 4.

Step 1: Substitute y in the First Equation

Given the second equation in the system, \(y = 2x\). This can be substituted directly into the first equation in place of y:\(x + (2x) = 6\).

Step 2: Solve for x

The equation from Step 1 simplifies to \(3x = 6\). Solving for x, it is found that \(x = 6 / 3 = 2\).

Step 3: Substitute x into Second Equation

Now substitute x = 2 into the second equation to solve for y. The equation, \(y = 2x\), turns to \(y = 2 * 2\).

Step 4: Solve for y

After substituting for x in the second equation, the equation now simply results in \(y = 4\).

Step 5: Check Proposed Solutions

The proposed solutions, x = 2 and y = 4, are checked by substituting them back into the original equations. For the first equation, it is found that \(2 + 4 = 6\). And for the second equation, \(4 = 2 * 2\) or \(4 = 4\) which is correct. Hence, the solutions x = 2, y = 4 are correct.

Key Concepts

Understanding Systems of Equations

When dealing with systems of equations, you're working with two or more equations that share common variables. The goal is to find values for these variables that satisfy all the equations simultaneously. In our example, the system is:

• \( x + y = 6 \)
• \( y = 2x \)

We're looking for values of \( x \) and \( y \) that work for both equations. Systems of equations can be solved using various methods, including substitution, elimination, or graphical methods. Here, we'll focus on the substitution method, which involves solving one equation for one variable, and then substituting that expression into another equation.

Exploring Algebra Concepts

Algebra is a branch of mathematics dealing with symbols and the rules for manipulating those symbols. In our system of equations, the primary algebraic skills used are substitution and basic arithmetic operations. The original equations establish a relationship between \( x \) and \( y \). From the second equation \( y = 2x \), we can express \( y \) in terms of \( x \), a key concept in substitution. This enables us to replace \( y \) with \( 2x \) in the first equation: \( x + 2x = 6 \). The algebraic manipulation simplifies the problem significantly, allowing for straightforward calculations and a clearer path to the solution.

Steps in Equation Solving Using Substitution

Solving equations requires a systematic approach to ensure accuracy. Here is how substitution works in practice:

• Step 1: Substitute - Use one equation to express one variable in terms of another. In our case, replace \( y \) with \( 2x \) in the first equation, leading to \( x + 2x = 6 \).
• Step 2: Solve for one variable - Simplify the equation to \( 3x = 6 \) and solve for \( x \) to find \( x = 2 \).
• Step 3: Substitute back - Use the found value to calculate the other variable. Using \( x = 2 \) in \( y = 2x \), results in \( y = 4 \).
• Step 4: Verify - Check your solution by substituting back into the original equations to ensure both are satisfied. For the first, \( 2 + 4 = 6 \) checks out, and for the second, \( 4 = 2 \times 2 \) is also correct.

This method is effective because it breaks down the problem into manageable parts, making it easier to avoid errors and understand the interaction between the two equations.