Problem 5 - Algebra A Combined Function

Question

Solve each system of equations by the substitution method. $$ \left\\{\begin{array}{l} y=3 x+1 \\ 4 y-8 x=12 \end{array}\right. $$

Step-by-Step Solution

Answer

The solution is \( (2, 7) \).

Step 1: Start with the given equations

We have two equations: 1) \( y = 3x + 1 \) 2) \( 4y - 8x = 12 \). The first equation is already solved for \( y \).

Step 2: Substitute y in the second equation

Replace \( y \) in the second equation with \( 3x + 1 \) from the first equation: \[ 4(3x + 1) - 8x = 12 \].

Step 3: Simplify the equation

Distribute the 4 in the equation: \[ 12x + 4 - 8x = 12 \].

Step 4: Combine like terms

Combine the \( x \) terms: \[ 4x + 4 = 12 \].

Step 5: Solve for x

Subtract 4 from both sides to isolate the \( x \) term: \[ 4x = 8 \]. Divide by 4: \[ x = 2 \].

Step 6: Substitute x back to find y

Substitute \( x = 2 \) back into the first equation: \( y = 3(2) + 1 = 6 + 1 = 7 \).

Step 7: State the solution

The solution to the system of equations is \( (x, y) = (2, 7) \).

Key Concepts

System of Equations

A system of equations is a collection of two or more equations with the same set of variables. In this exercise, the system consists of two equations with variables \( x \) and \( y \):

• \( y = 3x + 1 \)
• \( 4y - 8x = 12 \)

The solution to a system of equations is the set of values that make all the equations true simultaneously. Here, we use the substitution method to find those values. Understanding how to handle systems of equations is essential for tackling problems that involve multiple variables and constraints.
The substitution method is particularly useful when one of the equations is already solved for one of the variables, making it easier to substitute and eliminate variables step by step. This method allows us to solve for one variable at a time, simplifying the solution process.

Algebra

Algebra involves working with symbols and using them to express problems and relationships between quantities. In this exercise, we apply algebraic techniques to manipulate and solve equations.
The substitution method falls under algebra since it involves rearranging and simplifying equations to find unknown variables. Here, you used algebraic expressions like \( 4(3x + 1) - 8x = 12 \) and simplified them to solve the problem.
Algebra is fundamental in laying the groundwork for more complex mathematical concepts such as calculus and linear algebra. Mastering algebraic manipulation is key to efficiently solving equations and understanding mathematical relationships.

Linear Equations

Linear equations are equations of the first degree, which means they contain variables raised to the power of one. The given system of equations consists of linear equations. These take the form of straight lines when graphed on a coordinate plane.
In the exercise, both equations represent lines:

• The first equation, \( y = 3x + 1 \), is already in the slope-intercept form \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept.
• The second equation, \( 4y - 8x = 12 \), can also be manipulated into a similar form through simplification.

Solving linear equations through methods like substitution involves finding the point where these two lines intersect, which provides the solution to the system.

Solution of Equations

The solution of equations refers to finding the exact values for variable(s) that satisfy all given equations. When dealing with a system of equations, the solution represents a shared point of intersection between the graphs of the equations involved.
In this exercise, after substituting and simplifying, you found \( x = 2 \) and \( y = 7 \), meaning (2, 7) is the point where the two lines intersect. This is the solution to the system.
Finding solutions requires careful manipulation to ensure the operations transition smoothly, reducing errors. Real-world problems often demand accurate solutions, as these reflect the outcomes necessary in engineering, economics, and scientific contexts.