Problem 56 - Algebra 2

Question

How would you test whether \((2,-2)\) is a solution of the system?

Step-by-Step Solution

Answer

To test the point (2,-2), substitute x with 2 and y with -2 in each equation of the system and evaluate. If both equations are true with these values, then (2,-2) is a solution to the system.

Step 1: Understand the Task

First, understand that to test whether the point (2,-2) is a solution of the system, we must plug these values into each equation of the system and see if they make the equations true.

Step 2: Plug in the Point Into the First Equation

Identify the first equation in the system and substitute x with 2 and y with -2 to determine if the equation is satisfied.

Step 3: Check the First Equation

Evaluate the first equation after substitution to verify if the left side equals the right side.

Step 4: Plug in the Point Into the Second Equation

Identify the second equation in the system and substitute x with 2 and y with -2 to determine if the equation is satisfied.

Step 5: Check the Second Equation

Evaluate the second equation after substitution to verify if the left side equals the right side.

Key Concepts

Substitution Method

The substitution method is a cornerstone in solving systems of equations, particularly when dealing with two equations involving two variables. This method involves solving one of the equations for one variable in terms of the other and then substituting this expression into the other equation.

For example, if we have a system with equations 1) \(y = 2x + 3\) and 2) \(y = x - 1\), we can solve equation 2) for \(y\), giving us \(y = x - 1\). We then substitute this into equation 1), resulting in \(x - 1 = 2x + 3\), which simplifies to an equation with one variable that we can solve easily.

Utilizing the substitution method often requires simplifying algebraic expressions and carefully performing algebraic operations to ensure accurate solutions. It is particularly useful when equations are easily rearranged to isolate a variable and can be extended beyond two-variable systems.

System of Equations

A system of equations is simply a set of two or more equations that share a common set of variables. The main goal when working with systems of equations is to find the value of the variables that satisfies all equations in the system simultaneously. Typically, systems are categorized by the number of equations and variables they contain.

There are different methods to solve these systems, including graphing, substitution, elimination, and matrix approaches. The system is considered consistent if there is at least one set of values for the variables that satisfies all equations; otherwise, it is inconsistent. For linear systems, the solutions can sometimes be visualized as the intersection points of lines or planes represented by the equations in a Cartesian coordinate system.

Algebraic Solution Verification

Verification is the final, crucial step in solving algebraic problems. To validate the solutions of a system of equations, each solution must be substituted back into the original equations to ensure that both sides of the equations are equal.

For instance, using our example system and the point \( (2,-2) \), we check the solution by plugging in \( x = 2 \) and \( y = -2 \) into each of the original equations. If the left-hand side (LHS) equals the right-hand side (RHS) after substitution for all given equations, then we've confirmed that the point is indeed a solution to the system. This process acts as a proof that our algebraic manipulations have led to a valid solution. Should the substitution not hold true for every equation, this indicates either an error in calculation or that the proposed point is not a solution to the system.