Problem 24 - A Graphical Approach to College Algebra

Question

Solve each system by substitution. $$\begin{aligned}&3 y=5 x+6\\\&x+y=2\end{aligned}$$

Step-by-Step Solution

Answer

The solution is \((0, 2)\).

Step 1: Solve for one variable

To use substitution, we need to solve one of the equations for one of the variables. Let's solve the second equation, \( x + y = 2 \), for \( y \). This gives us \( y = 2 - x \).

Step 2: Substitute into the other equation

Now, take the expression for \( y \) which is \( y = 2 - x \) and substitute it into the first equation. Replace \( y \) in the equation \( 3y = 5x + 6 \) to get: \( 3(2 - x) = 5x + 6 \).

Step 3: Simplify and solve for \( x \)

Distribute the 3 on the left-hand side: \( 6 - 3x = 5x + 6 \). Next, rearrange the terms to get all the \( x \) terms on one side: \( 6 - 6 = 5x + 3x \), simplifying gives \( 0 = 8x \). Divide by 8 to find \( x = 0 \).

Step 4: Substitute back to find \( y \)

Use the value of \( x \) to find \( y \). Substitute \( x = 0 \) into the equation \( y = 2 - x \), thus \( y = 2 - 0 \), which simplifies to \( y = 2 \).

Step 5: Write the solution

The solution to the system is \( x = 0 \) and \( y = 2 \). So, the solution is the ordered pair \( (0, 2) \).

Key Concepts

Substitution Method

The substitution method is a powerful technique used to solve systems of equations, particularly when one of the equations is easily solvable for one variable. By solving one equation for a variable, we can express this variable in terms of the other, allowing us to substitute it into the second equation.
For example, consider the system of equations given by:

• \(3y = 5x + 6\)
• \(x + y = 2\)

To use the substitution method, we start by solving one of these equations for one of the variables. Here, the second equation \(x + y = 2\) is easily solved for \(y\), giving \(y = 2 - x\).
This expression for \(y\) can then replace \(y\) in the first equation, leading to a new equation in terms of just one variable, \(x\), simplifying the process of solving the system step by step.

Linear Equations

Linear equations form the basis of many algebraic solutions and are simply equations involving variables raised to the power of one. This means there are no squared terms or higher-powered terms involved. The general form of a linear equation is \(ax + by = c\), where \(a\), \(b\), and \(c\) are constants.
In our exercise, we have two linear equations:

• \(3y = 5x + 6\)
• \(x + y = 2\)

Both equations involve variables \(x\) and \(y\) whose degree is one, classifying them as linear. These types of equations are straightforward to manipulate, as they allow for simple algebraic operations like addition, subtraction, and multiplication to find the values of \(x\) and \(y\).
Linear equations often appear in many real-world scenarios such as calculating distances, managing budgets, or modeling business profits and expenses, making them very practical as well.

Algebraic Solution Steps

Solving equations through algebraic steps involves systematic manipulation of the equations to isolate the variables and find their values. For this particular exercise, follow these steps to solve for \(x\) and \(y\) using substitution:

• **Step 1:** Solve one equation for a single variable, for instance, using \(x + y = 2\), solve for \(y\) giving \(y = 2 - x\).
• **Step 2:** Substitute this expression for \(y\) back into the other equation \(3y = 5x + 6\). This gives a single-variable equation \(3(2-x) = 5x + 6\).
• **Step 3:** Simplify this new equation using basic algebraic operations, like distributing the 3: \(6 - 3x = 5x + 6\).
• **Step 4:** Rearrange the equation to collect all \(x\) terms on one side, resulting in \(0 = 8x\).
• **Step 5:** Solve for \(x\) by dividing both sides by 8, giving \(x = 0\).
• **Step 6:** Use this value of \(x\) in the expression for \(y\) to find \(y = 2\).
• **Step 7:** Write the solution as the ordered pair \((0, 2)\), verifying it satisfies both original equations.

This structured approach ensures a logical and comprehensive way to solve systems of equations, minimizing errors while maximizing understanding.