Problem 1 - Algebra 2

Question

Solve each system of equations by using substitution. \(y=3 x-4\) \(y=4+x\)

Step-by-Step Solution

Answer

The solution is (4, 8).

Step 1: Write Both Equations

We are given two equations in the system:1. \( y = 3x - 4 \)2. \( y = x + 4 \)

Step 2: Set Equations Equal to Each Other

Since both equations are equal to \( y \), we can set them equal to each other:\[ 3x - 4 = x + 4 \]

Step 3: Solve for x

First, subtract \( x \) from both sides of the equation:\[ 3x - x - 4 = 4 \]Which simplifies to:\[ 2x - 4 = 4 \]Next, add \( 4 \) to both sides:\[ 2x = 8 \]Finally, divide by 2 to solve for \( x \):\[ x = 4 \]

Step 4: Substitute x Back into One of the Original Equations

Now that we have \( x = 4 \), substitute this value back into one of the original equations, such as \( y = x + 4 \):\[ y = 4 + 4 \]Therefore, \( y = 8 \).

Step 5: Write the Solution as an Ordered Pair

The solution to the system of equations is the ordered pair \( (x, y) \). From the previous steps, we found that \( x = 4 \) and \( y = 8 \). So, the solution is \( (4, 8) \).

Key Concepts

Substitution Method

The substitution method is a technique for solving systems of equations. By solving one equation for one variable, you can substitute this expression into the other equation. This method focuses on making one variable the subject of one equation, which simplifies solving the system. Consider two equations:

• First equation: \( y = 3x - 4 \)
• Second equation: \( y = x + 4 \)

These expressions both represent \( y \). By equating them, you simplify down to one equation with one variable: \( 3x - 4 = x + 4 \). This conversion is the crux of the substitution method. It makes finding the solution straightforward once you only treat one variable at a time.

Solving Linear Equations

In mathematics, solving linear equations refers to finding the value of the unknown variable that makes the equation true. Linear equations are of the form \( ax + b = c \). Solving such equations often involves isolating the variable \( x \) by performing operations such as addition, subtraction, multiplication, or division.From our example, after substituting and simplifying, you end up with \( 2x = 8 \). To solve for \( x \), divide both sides by 2. This process steps through:

• Subtract the smaller variable expression from both sides to collect all \( x \)-terms on one side.
• Balance the equation by performing inverse operations, like adding or subtracting the same number from both sides.
• Finally, isolate the variable by dividing by the coefficient of \( x \).

Solving linear equations is foundational in algebra and helps unlock complex systems by breaking them down into simpler tasks.

Algebraic Manipulation

Algebraic manipulation involves rearranging and simplifying equations to make them easier to solve. It's a skill used to transform equations, keeping them balanced while isolating variables or combining terms. Successful manipulation requires adhering to algebraic identities and operations rules.In our context, algebraic manipulation starts when we substitute and equate two expressions for \( y \), giving \( 3x - 4 = x + 4 \). The process here includes:

• Subtracting \( x \) from both sides: \( 3x - x \)
• Simplifying to combine like terms: \( 2x - 4 \)
• Continuing by adding the opposite operation to both sides to isolate terms.

This manipulation is key to solving variables step-by-step, maintaining the integrity of the equations while simplifying them for clarity and solution.