Problem 4 - Intermediate Algebra

Question

To solve \(\left\\{\begin{array}{l}y=3 x+1 \\ x+y=4\end{array}\right.\) we can ____________ \(3 x+1\) for \(y\) in the second equation.

Step-by-Step Solution

Answer

The solution is \( x = \frac{3}{4} \) and \( y = \frac{13}{4} \).

Step 1: Substitute the Expression for y

The system of equations is: \( y = 3x + 1 \) and \( x + y = 4 \). Substitute \( y = 3x + 1 \) into the second equation to replace \( y \). This gives us: \[ x + (3x + 1) = 4 \].

Step 2: Combine Like Terms

Combine like terms in the equation \( x + 3x + 1 = 4 \). This simplifies to \[ 4x + 1 = 4 \].

Step 3: Solve for x

To isolate \( x \), subtract 1 from both sides of the equation: \[ 4x + 1 - 1 = 4 - 1 \], which simplifies to \[ 4x = 3 \]. Then, divide both sides by 4: \[ x = \frac{3}{4} \].

Step 4: Substitute Back to Find y

Now that we have \( x \), substitute \( x = \frac{3}{4} \) back into the equation \( y = 3x + 1 \) to find \( y \). This gives us: \[ y = 3 \left( \frac{3}{4} \right) + 1 \].

Step 5: Simplify to Find y

Calculate \( 3 \times \frac{3}{4} \): \( \frac{9}{4} \). Add 1 (or \( \frac{4}{4} \)) to \( \frac{9}{4} \) to get \[ y = \frac{9}{4} + \frac{4}{4} = \frac{13}{4} \].

Key Concepts

Substitution Method

The substitution method is a systematic way to solve a system of linear equations. This method involves solving one equation for one variable and then substituting that expression into the other equation.
To illustrate, consider our system of equations:

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

In this example, Equation 1 has already solved for \( y \). We see that \( y \) is expressed in terms of \( x \) as \( y = 3x + 1 \).
This expression can be substituted directly into the second equation, replacing \( y \) with \( 3x + 1 \). This results in one equation with one variable: \( x + (3x + 1) = 4 \).
By performing this substitution, we effectively reduce the system from two equations down to one, making it much easier to solve the unknown variable. This step is crucial for simplifying the overall problem.

Combining Like Terms

Once the substitution is complete, you often face an equation with similar terms on one side. Being able to identify and combine like terms is a fundamental skill in algebra that simplifies such equations.
In our substituted equation:

• \( x + 3x + 1 = 4 \)

The terms \( x \) and \( 3x \) are similar since they contain the same variable \( x \). Combining them results in \( 4x \).
This action simplifies the equation to \( 4x + 1 = 4 \), allowing for easier manipulation in the next steps.
Combining like terms efficiently reduces complexity, preventing errors and simplifying the path to finding the solution. Make sure to always double-check that all similar terms are correctly combined before moving on.

Solving for Variables

After you've combined like terms, the next step is to isolate the variable in the equation. This process is known as solving for the variable. For our specific problem, we ended with the equation \( 4x + 1 = 4 \).
To isolate \( x \), we need to get rid of any numbers that are added or multiplied by it. The strategy generally involves

• Reversing addition or subtraction
• Reversing multiplication or division

Start by subtracting 1 from both sides, leaving us with \( 4x = 3 \).
Then, divide both sides by 4 to solve for \( x \):

• \( x = \frac{3}{4} \)

Once \( x \) is found, we can substitute it back into the original "solved-for" expression for \( y \), or any initial equation containing \( y \), to find its value.
Substitute \( x = \frac{3}{4} \) in \( y = 3x + 1 \) and solve:

• \( y = 3\left(\frac{3}{4}\right) + 1 = \frac{9}{4} + \frac{4}{4} = \frac{13}{4} \)

By directly tackling equations one step at a time, solving for variables turns into a straightforward, logical process.