Problem 3 - Algebra Form and Function

Question

Solve the systems of equations. $$ \left\\{\begin{array}{l} 3 x-4 y=7 \\ y=4 x-5 \end{array}\right. $$

Step-by-Step Solution

Answer

Question: Find the coordinates (x, y) where the following system of equations is true: \(3x - 4y = 7\) \(y = 4x - 5\) Answer: The solution to the given system of equations is (x, y) = (1, -1).

Step 1: Substitute y in the first equation

We will start by substituting the expression for y from the second equation (\(y = 4x - 5\)) into the first equation (\(3x - 4y = 7\)). This substitution gives: $$ 3x - 4(4x - 5) = 7 $$

Step 2: Solve for x

Now, let's distribute the -4 through the parentheses and consolidate all x terms on one side: $$ 3x - 16x + 20 = 7 $$ Combine the terms with x, getting: $$ -13x + 20 = 7 $$ Now subtract 20 from both sides: $$ -13x = -13 $$ To find the value of x, divide both sides by -13: $$ x = 1 $$

Step 3: Find the value of y

Now that we have the value of x, we can substitute it back into the expression for y from the second equation: $$ y = 4(1) - 5 $$ Multiply and simplify: $$ y = 4 - 5 = -1 $$

Step 4: Write the final answer

The solution to the system of equations is the point (x, y) where both equations are true. Thus, the solution is: $$(x, y) = (1, -1)$$

Key Concepts

Substitution Method

Substitution is a powerful technique for solving systems of equations, especially when one equation is already solved for one variable. This method involves replacing one variable in an equation with an expression derived from another equation. It simplifies the process by reducing the number of variables temporarily so you can more easily solve for one of them.

In the given problem, we have a system of equations:

• First equation: \( 3x - 4y = 7 \)
• Second equation: \( y = 4x - 5 \)

To use the substitution method, we start by taking the already solved equation for \( y \), \( y = 4x - 5 \), and substitute it into the first equation. This substitution changes the first equation into one that only involves \( x \), allowing us to solve for \( x \). Once \( x \) is found, we can substitute back to find \( y \), completing the solution.

Solution of Equations

Solving equations is essentially the process of finding values for unknown variables that make the equations true. In a system like the one we're solving, you deal with two equations simultaneously. The solution is the point where both equations are satisfied—meaning both equations are true at the same time.

After substituting \( y \) with \( 4x - 5 \), the first equation becomes: \( 3x - 4(4x - 5) = 7 \). This results in a one-variable equation. By solving this equation, we pinpoint the exact value of \( x \) that, when used in both equations, will make them both true. After finding \( x \), we substitute it back into either of the original equations to find \( y \). Together, these values determine the point of intersection in the coordinate plane, solving the system of equations.

Algebraic Manipulation

Algebraic manipulation involves techniques such as distributing, combining like terms, and using inverse operations to solve equations. These are essential skills for isolating and solving for variables.

Once we substituted \( y = 4x - 5 \) into the first equation, the next step was to distribute the \(-4\) across \( (4x - 5) \). This gives us \( 3x - 16x + 20 = 7 \). By combining the \( x \)-terms, we simplify the equation to \( -13x + 20 = 7 \).

Subtracting 20 from both sides further simplifies to \( -13x = -13 \). Finally, by dividing both sides by \(-13\), we isolate \( x \) to get \( x = 1 \). After finding \( x \), substituting back into the equation for \( y \) involves a simple multiplication and subtraction operation: \( y = 4(1) - 5 = -1 \).

These algebraic maneuvers are the stepping stones to solving equations efficiently and understanding the relationships between variables.