Question

\(x=\frac{5}{3}, y=-7\)

Short answer

The provided values are the solutions: \(x = \frac{5}{3}\), \(y = -7\)

Step-by-step solution

Understanding the Given Information

The values for \(x\) and \(y\) have been provided directly. Since this isn't a problem that requires solving a system of equations or any other mathematical issue, there are no intermediate steps to take. \(x\) is defined as being equal to \(\frac{5}{3}\) and \(y\) as being equal to \(-7\). This is already the solution.

Key concepts

Systems of Equations

A system of equations consists of two or more equations with different variables that you solve at the same time. The goal is to find a common solution that satisfies all the equations in the system. These can be linear equations, like the ones you might see in simple algebra classes, or more complex forms, such as non-linear systems that you'd encounter in advanced mathematics.

There are different methods to solve these systems, including graphing, substitution, elimination, and matrix operations. Graphing involves plotting the equations on a graph to see where they intersect, but this method can be imprecise. Algebraic methods like substitution and elimination are more accurate, especially when dealing with linear systems.

Algebraic Solutions

An algebraic solution is a process of finding the values of variables that satisfy the terms of an algebraic equation. In the context of a system of equations, you'll have to find the values for each variable that make all the equations true simultaneously.

For individual equations, an algebraic solution involves manipulating the equation to isolate the variable you're solving for, often through operations like addition, subtraction, multiplication, division, or factoring. The exercise provided doesn't call for these steps, as it already presents the solved values of the variables, bypassing the need for algebraic manipulation.

Substitution Method

The substitution method is a technique used to find solutions to systems of equations. You start by solving one of the equations for one variable in terms of the other(s). Then, you substitute this expression into another equation, effectively reducing the number of variables and equations.

This process is repeated until you solve for one variable and then back-substitute the known values to find the others. While the provided exercise doesn't require you to employ this method since the solutions are given, understanding substitution is key for solving similar problems that don't offer variables' values outright. It helps in transitioning from complex systems to simpler, more manageable single-variable equations.