Problem 4 - Algebra A Combined Function

Question

Solve each system of equations by the substitution method. $$ \left\\{\begin{array}{l} x+y=6 \\ y=-4 x \end{array}\right. $$

Step-by-Step Solution

Answer

The solution is \( x = -2 \) and \( y = 8 \).

Step 1: Identify the Equations

The given system of equations is:\[\begin{align*}1) & \quad x + y = 6 \2) & \quad y = -4x\end{align*}\]

Step 2: Substitute for y

Substitute the expression for \( y \) from the second equation into the first equation. We substitute \( y = -4x \) into \( x + y = 6 \):\[ x + (-4x) = 6 \]

Step 3: Simplify and Solve for x

Simplify the equation from Step 2:\[ x - 4x = 6 \]which simplifies to:\[ -3x = 6 \]Divide both sides by \(-3\) to solve for \( x \):\[ x = -2 \]

Step 4: Substitute x back to Find y

Now, substitute \( x = -2 \) into the second equation to find \( y \):\[ y = -4(-2) \]Calculate the result:\[ y = 8 \]

Step 5: Solution Verification

Verify the solution by substituting \( x = -2 \) and \( y = 8 \) back into the first equation:\[ x + y = 6 \Rightarrow -2 + 8 = 6 \]Since the equation holds true, our solution is verified.

Key Concepts

System of Equations

A system of equations is a set of two or more equations that involve the same variables. In this problem, our particular system involves two equations with the variables, \( x \) and \( y \): \( x + y = 6 \) and \( y = -4x \). These types of problem sets are common in algebra because they challenge you to find values for the unknowns that hold true for all equations involved.

Whenever you approach a system of equations, your goal is to find the values that satisfy every equation simultaneously. This is often visualized as finding the intersection point of two lines if you were to graph the equations. In essence, the solution to the system of equations is the coordinate or coordinates that fit perfectly into both equations.

• The number of solutions can vary depending on the system. Some systems may have one solution, many solutions, or even no solutions at all.
• Techniques to solve systems of equations include substitution, elimination, and graphical methods.

Substitution, the method used in our example, is ideal when at least one of the equations is solved for one of the variables. It involves replacing one variable with an equivalent expression in terms of the other variable.

Algebra

Algebra is a branch of mathematics that deals with symbols and the rules for manipulating those symbols to solve equations or study relationships. When you work with algebra, you transform everyday descriptions of problems into mathematical equations. This allows us to systematically find solutions for various scenarios.

In this exercise, algebraic skills are used to manipulate the original system of equations to find the values of \( x \) and \( y \). This involves using basic operations like addition, subtraction, and division to isolate variables and simplify the equations.

• The power of algebra lies in its ability to model real-world situations with equations and to find solutions that may not be immediately visible.
• It enables us to understand and describe complex patterns and relationships succinctly.

Learning how to effectively use algebra helps in building strong problem-solving skills which are applicable in numerous fields beyond mathematics itself.

Solving Equations

Solving equations is the process of finding out what values the variables in the equations represent. In this exercise, we use substitution to solve the system of equations. This involves finding expressions for the variables from one equation and substituting them into the other.

Let's look at how substitution works:

• First, express one variable in terms of others using one of the equations. In our problem, we have \( y = -4x \), so \( y \) is already expressed in terms of \( x \).
• Next, replace \( y \) in the other equation with \(-4x\), leading to a single equation with a single variable.
• Solve for that variable. We get \( x = -2 \) by simplifying and solving the equation \( x - 4x = 6 \).
• Substitute this value back into one of the original equations to find the corresponding value of the other variable. Plugging \( x = -2 \) into \( y = -4x \) gives \( y = 8 \).

Finally, it is crucial to verify the final results by substituting both \( x \) and \( y \) values back into the original equations. This ensures that the solution is consistent and accurate within the context of each equation present in the system.