Problem 5 - College Algebra

Question

In Exercises \(5-18\), solve each system by the substitution method. $$ \begin{aligned} &x+y=4\\\ &y=3 x \end{aligned} $$

Step-by-Step Solution

Answer

(x, y) = (1, 3)

Step 1: Substitution

Replace \(y\) in the first equation with \(3x\) from the second equation. This gives us the equation \(x + 3x = 4\).

Step 2: Solve for x

Combine like terms on the left side of the equation to get \(4x = 4\). Then, divide each side of the equation by 4 to isolate x, which gives us the solution \(x = 1\).

Step 3: Solve for y

Substitute \(x = 1\) into the second equation, yielding \(y = 3*1\). Therefore, \(y = 3\).

Key Concepts

System of Equations

A system of equations is a set of two or more equations that have common variables. In our problem, we have two linear equations:

• \(x + y = 4\)
• \(y = 3x\)

These equations are connected by the variables \(x\) and \(y\), and our task is to find values for these variables that satisfy both equations at the same time.
When dealing with a system of equations, the solutions represent the points where the equations intersect if we were to graph them. In this context, intersection means that the values of \(x\) and \(y\) make both equations true simultaneously.
Systems of equations can be solved using various methods like substitution, elimination, and graphical methods. Here, we are focusing on the substitution method.

Solving Equations

Solving equations involves finding the values of the unknown variables that make the equation true. For any algebraic equation, our goal is to manipulate the equation so that the variable is isolated on one side.
Starting with the given system, solve for one variable in terms of the other using the substitution method. For instance, from \(y = 3x\), we already have \(y\) expressed in terms of \(x\). This makes one step of solving easier because we can replace \(y\) in other equations.
The solution requires basic operations, such as:

• Combining like terms
• Isolating the variable through addition, subtraction, multiplication, or division

By substituting \(3x\) for \(y\) in \(x+y=4\), we simplify to a single equation with one variable, making it solvable using basic algebraic techniques.

Algebraic Substitution

Algebraic substitution is a method where one variable in an equation is replaced by an expression derived from another equation. This approach helps simplify systems of equations, reducing the number of unknowns in each equation.
For our problem:

• The second equation \(y = 3x\) serves as an expression for \(y\).
• We substitute \(3x\) in place of \(y\) in the first equation \(x + y = 4\).

This conversion creates a simpler equation, \(x + 3x = 4\), which can be further simplified to \(4x = 4\). Solving this results in \(x = 1\).
Now we have a value for \(x\), allowing us to backtrack and find \(y\) using the same substitution method, giving \(y = 3 * 1 = 3\). Thus, substitution efficiently reduces complexity, allowing for easier solutions.

Steps in Problem Solving

Problem solving using substitution generally involves strategic steps that guide us toward a solution:

• Identify the equations in the system.
• Solve one of the equations for one variable, if it isn’t already.
• Substitute the expression found into the other equation.
• Simplify and solve the resulting equation for the single variable.
• Use the value obtained to back-substitute and find the other variable.

In our example, the substitution method simplified the process by effectively converting a system of two equations into a single linear equation. Breaking the process down into specific steps clarifies the pathway to the solution and ensures no steps are skipped.
This method not only offers analytical clarity but also boosts confidence in solving future equations because of its orderly nature.