Problem 1 - Elementary Algebra Within Reach

Question

In Exercises 1-4, solve the system by the method of substitution. $$ \left\\{\begin{array}{l} y=2 x-1 \\ y=-x+5 \end{array}\right. $$

Step-by-Step Solution

Answer

The solution to the system of equations is \(x = 2\) and \(y = 3\).

Step 1: Apply the Substitution Method

The method of substitution involves substituting one equation into another. We have two equations here: \(y = 2x - 1\) and \(y = -x + 5\). Replace \(y\) in the second equation with \(2x - 1\) since \(y = 2x - 1\).

Step 2: Solve for x

The second equation \(y = -x + 5\) becomes \(2x - 1 = -x + 5\). Solve this equation for \(x\) by adding \(x\) to both sides of the equation which yields \(3x - 1 = 5\). Now add 1 to both sides, we get \(3x = 6\). Dividing both sides by 3, gives us the solution \(x = 2\).

Step 3: Solve for y

Substitute \(x = 2\) into either equation to solve for \(y\). For simplicity, let's substitute it into the first equation, which gives us \(y = 2(2) - 1 = 3\).

Key Concepts

Method of Substitution

The method of substitution is a common technique used to solve systems of equations, especially when dealing with linear equations. The basic idea is to solve one of the equations for one variable, and then substitute that expression into the other equation. This process simplifies the system of equations into a single equation with one variable.

Here’s why substitution is useful:

• It reduces the number of equations and variables, simplifying the solution process.
• It's particularly effective when one equation is already solved for a single variable, or can be easily manipulated to get one.
• Once a variable is isolated, it eliminates that variable from subsequent equations, streamlining the solution.

By substituting one equation into another, it allows us to solve complex systems by breaking down the problem into smaller, more manageable parts. This method is generally best suited for systems of linear equations, but can also be applied to more complex nonlinear systems with additional techniques.

Solving Linear Equations

Solving linear equations effectively requires understanding the properties of equality and how to manipulate equations to isolate variables. When given a linear equation, the goal is to find the value of the unknown variable that makes the equation true. The steps generally include:

• Combining like terms on both sides of the equation wherever possible.
• Using addition, subtraction, multiplication, or division to isolate the variable.
• Keeping the equation balanced by performing the same operation on both sides.

This process helps solve equations by systematically eliminating all operations surrounding the variable, thereby finding its value.

For example, with a simple equation like \(2x - 1 = 5\), one might isolate \(x\) by first adding 1 to both sides (\(2x = 6\)), and then dividing each side by 2, yielding \(x = 3\). This kind of careful manipulation ensures that all possible solutions are discovered for the variable in the context of the equation given.

Substitution Method Example

Let's look at an example to see the substitution method in action, applied to solving a system of linear equations:

Consider the system:

• Equation 1: \(y = 2x - 1\)
• Equation 2: \(y = -x + 5\)

First, notice that Equation 1 already has \(y\) isolated. We can use this expression for \(y\) from Equation 1 and substitute it directly into Equation 2. This gives us:

\(2x - 1 = -x + 5\)

Now, solve for \(x\):
- Add \(x\) to both sides: \(3x - 1 = 5\)
- Add 1 to both sides: \(3x = 6\)
- Divide by 3: \(x = 2\)

Next, substitute \(x = 2\) back into Equation 1 to find \(y\):
- \(y = 2(2) - 1 = 3\)

Thus, the solution to the system is \((x, y) = (2, 3)\). This example clearly illustrates how the substitution method can simplify finding solutions to systems of equations by handling one variable at a time.