Problem 5 - College Algebra Essentials

Question

In Exercises \(5-18,\) solve each system by the substitution method. \(\left\\{\begin{array}{l}x+y=4 \\ y=3 x\end{array}\right.\)

Step-by-Step Solution

Answer

The solution to the system of equations is \(x = 1\) and \(y = 3\). In ordered pair format, the solution is (1,3).

Step 1 : Substitute the equation

Solve the system by the substitution method. Substitute \(y = 3x\) from the second equation into the first equation to get: \(x + 3x = 4\), which simplifies to \(4x = 4\).

Step 2 : Solve for \(x\)

Divide both sides of the equation by 4 to solve for \(x\): \(x = 4/4 = 1\). This is the value for \(x\).

Step 3 : Solve for \(y\)

Substitute \(x = 1\) into the second equation \(y = 3x\) and simplify to find the corresponding value for \(y\): \(y = 3*1 = 3\).

Step 4 : Write the solution as an ordered pair

The solution to the system of equations is the ordered pair (1,3).

Key Concepts

System of Equations

When we talk about a system of equations, we're dealing with more than one equation at the same time. A system of equations can help us find out where two lines intersect on a graph.
In this exercise, we have two equations:

• Equation 1: \(x + y = 4\)
• Equation 2: \(y = 3x\)

Our goal is to find a set of values for \(x\) and \(y\) that make both equations true. This involves finding the point at which the two lines represented by these equations intersect. Such a point is called the solution to the system. In this particular exercise, solving it will give us the exact values for \(x\) and \(y\) where these two lines cross.
For more complex systems of equations, there might be no solutions or even infinite solutions. But in our example, we're lucky because there is exactly one solution.

Algebra

Algebra is really all about using letters to represent numbers, allowing us to solve equations and find unknown values.
When dealing with the substitution method, we're demonstrating one of algebra's fundamental skills—manipulating equations to discover the value of variables. In our example, we're using the substitution method to solve the system of equations, which is a common algebraic solution technique. The key steps include:

• Identifying the two equations we have and recognizing which variable is already isolated, or when it seems easy to isolate one.
• Substituting the expression we find for that isolated variable into the other equation, simplifying as needed.

By doing this, we reduce two equations with two variables down to a single equation containing just one variable. This simplification helps us focus on finding the value for one variable at a time. Through this example, we see how algebra uses known values and operations to unlock solutions to equations and systems effectively.

Solving Equations

Solving equations is like unlocking a puzzle to find unknowns. With each step, you aim to move closer to getting the variable alone on one side of the equation.
In the substitution method, we use one equation to solve for one variable and then substitute it into another equation.
This process follows:

• Substitute: We first substitute \(y = 3x\) from the second equation into the first, simplifying our problem and replacing all \(y\)'s with \(3x\).
• Simplify: After substitution, we simplify the resulting equation—\(x + 3x = 4\) becomes \(4x = 4\).
• Solve: By dividing both sides by 4, we find \(x = 1\). Then substitute \(x = 1\) back into \(y = 3x\) to get \(y = 3\).

This technique systematically breaks down the problem, making it easier to solve each variable one at a time.
The solution to the system is the point where the equations "meet" on the graph, here represented as the ordered pair \((1, 3)\). This ordered pair directly represents the values of \(x\) and \(y\) in our system, confirmed through the process of solving these equations.