Problem 6 - College Algebra

Question

In Exercises \(5-18\), solve each system by the substitution method. $$ \begin{aligned} &x+y=6\\\ &y=2 x \end{aligned} $$

Step-by-Step Solution

Answer

The solution for the system of equations \(x+y=6\) and \(y=2x\) is \(x=2\) and \(y=4\).

Step 1: Substitute y in First Equation

The second equation is \(y=2x\). Substitute this into the first equation, i.e., replace \(y\) in \(x+y=6\), we get: \(x + 2x = 6\)

Step 2: Solve the Equation after Substitution

Now, solve the equation after substitution. Add the \(x\) terms together we get \(3x=6\). Solve for \(x\) and the value of \(x\) is \(x=2\).

Step 3: Substitute x in Second Equation

Having obtained the value of \(x\), we substitute \(x=2\) into the second equation \(y=2x\) to determine the value of \(y\). It gives us that \(y=2*2\), so \(y=4\)

Key Concepts

Substitution Method

The substitution method is an algebraic technique commonly used to solve systems of equations. This method involves replacing one variable with an expression involving the other variable, making it possible to solve for one unknown at a time.

Here's how to apply the substitution method step by step:

• First, isolate one of the variables in one of the equations.
• Next, substitute the expression for this variable into the other equation.
• Solve the resulting equation for the remaining variable.
• Finally, replace the found value into one of the original equations to solve for the other variable.

Using this method, you transform a system of equations into a single-variable problem which can be easily solved.

Algebraic Equations

Algebraic equations are mathematical statements that express the equality between two algebraic expressions. These equations have variables which are unknown quantities and can take various forms such as linear, quadratic, or polynomial.

The key to solving algebraic equations is to isolate the variable on one side of the equation. This can be achieved through operations such as addition, subtraction, multiplication, and division, applied to both sides of the equation. It's crucial to maintain the balance of the equation, meaning whatever you do to one side, you must do to the other side.

Importance of Order of Operations

Remembering the order of operations is essential when manipulating equations. The acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction) can help you remember the correct sequence.

Linear Systems

Linear systems consist of two or more linear equations involving the same set of variables. A solution to a linear system is an assignment of values to the variables that makes all the equations hold true simultaneously.

These systems can have:

• A single solution (where lines intersect at a point).
• No solution (where lines are parallel and never intersect).
• Infinitely many solutions (where lines coincide).

Understanding the structure of linear systems is foundational for algebra. They model many real-world situations where we need to find the point of equilibrium or the point where two different criteria meet. The substitution method is just one technique to solve these systems; others include the elimination method and graphing.