Problem 15 - Algebra 2

Question

Solve each equation. $$ \left[\begin{array}{ll}{4 x} & {3 y}\end{array}\right]=\left[\begin{array}{ll}{12} & {-1}\end{array}\right] $$

Step-by-Step Solution

Answer

x = 3, y = -1/3.

Step 1: Understanding the Problem

We are given two equations in matrix form: \( [4x, 3y] = [12, -1] \). This means that \( 4x = 12 \) and \( 3y = -1 \). We must solve each of these equations separately for \( x \) and \( y \).

Step 2: Solving for x

The equation to solve is \( 4x = 12 \). To isolate \( x \), divide both sides by 4: \[ x = \frac{12}{4} \]. Simplifying gives \( x = 3 \).

Step 3: Solving for y

Next, we solve the equation \( 3y = -1 \). Divide both sides by 3 to get \( y \): \[ y = \frac{-1}{3} \]. Thus, \( y = -\frac{1}{3} \).

Step 4: Solution Summary

We have solved the two equations independently. The solutions are \( x = 3 \) and \( y = -\frac{1}{3} \).

Key Concepts

Matrix Representation

A matrix is a compact and organized way to display multiple equations or data sets neatly. In the problem you encountered, the matrix \([4x, 3y] = [12, -1]\) is used to represent two separate equations without writing each one down in traditional format. This representation makes it easier to visualize and manage especially when working with more than two equations or variables.

• The first set of numbers \([4x, 3y]\) represents the coefficients of the equations.


• The second set \([12, -1]\) represents the solutions or outcomes of those particular equations.

By understanding this arrangement, you can see that each entry within the matrix corresponds to an equation: \(4x = 12\) and \(3y = -1\). Breaking them down in this way allows for simpler computation and solution finding.

Isolating Variables

Isolating variables is a fundamental step in solving equations. It involves rearranging the equation so that the variable of interest is on one side, isolated from other numbers or variables. This gives a clear path to find the solution for the unknown.

• For the equation \(4x = 12\), we isolate \(x\) by dividing both sides by 4, following the principle of maintaining balance in the equation.


• This results in \(x = \frac{12}{4}\) which simplifies to \(x = 3\).


• Similarly, for \(3y = -1\), divide both sides by 3 to isolate \(y\), leading to \(y = \frac{-1}{3}\).

The idea is to perform the same operation on both sides of the equation, ensuring that the equality is maintained while simplifying the problem.

Simplifying Fractions

Simplifying fractions is a crucial skill that makes equations much more manageable and results easier to interpret. When solving the equations provided, you'll encounter fractions that can be simplified.In the step \(x = \frac{12}{4}\), the simplification process involves dividing the numerator and the denominator by their greatest common factor, which is 4 in this case:\[x = \frac{12 \div 4}{4 \div 4} = \frac{3}{1} = 3\]For \(y = \frac{-1}{3}\), the fraction is already in its simplest form as -1 and 3 have no common factors other than 1, so it stays as is.Knowing how to effectively simplify fractions will always lead to a cleaner and often more intuitive result.

Independent Equations

When dealing with a set of linear equations, they are considered independent if their solution solely depends on their unique set of coefficients and constants. This means each equation can be solved on its own without needing information from the other.In the given matrix representation, you have two independent equations:

• \(4x = 12\), which is solved for \(x = 3\)


• \(3y = -1\), which is solved for \(y = -\frac{1}{3}\)

Since neither equation relies on the other for finding its solution, they demonstrate complete independence. Understanding the concept of independent equations helps in logically approaching and solving systems with multiple variables.