Problem 7 - Elementary Algebra Within Reach

Question

In Exercises 5-14, solve the system by the method of substitution. $$ \left\\{\begin{array}{r} 2 x=8 \\ x-2 y=12 \end{array}\right. $$

Step-by-Step Solution

Answer

The solution to the system of equations is x = 4, y = -4.

Step 1: Simplify first equation

First take the equation \(2x = 8\) and solve for x, by dividing both sides by 2: \(x = 8 / 2\). So, \(x = 4\)

Step 2: Substitute x in second equation

Now that we have x = 4, we can substitute this value into the second equation, which is \(x - 2y = 12\). Substituting, we get \(4 - 2y = 12\). Arranging the equation for y, we have \(-2y = 12 - 4\), which simplifies to \(-2y = 8\)

Step 3: Solve for y

Now we can solve for y by dividing both sides by -2. Therefore, \(y = 8 / -2\). In this final step, we find that \(y = -4\).

Key Concepts

Solving Systems of Equations

A system of equations is a set of two or more equations that you deal with at once, finding a common solution. Each equation in the system represents a line on a graph. The solution to the system is the point (or points) where the lines intersect. Solving systems of equations can help us find values that satisfy all equations involved. This intersection point is the common solution that makes all the equations in the system true simultaneously.

There are several methods to solve these systems, including graphing, substitution, and elimination. In this exercise, we focus on the substitution method, which is especially handy when one of the equations is already solved for one variable, or can be easily solved for one variable.

• Start by isolating one variable in one of the equations.
• Substitute the expression for this variable into the other equation.
• Solve the resulting equation to find the value of one variable.
• Substitute back to find the other variable.

This systematic approach ensures that all solutions obtained are valid for both equations.

Algebra Problems

Algebra can initially be challenging as it involves both symbols and numbers, which can seem abstract. However, algebra equips us with a powerful tool to solve real-world problems by allowing numerical relationships to be expressed in symbolic form. In our system of equations example, we convert a word problem or a scenario into algebraic expressions and then solve them to find solutions.

When tackling algebra problems like solving systems of equations, we often rely on basic algebraic operations such as addition, subtraction, multiplication, and division. Being comfortable with these operations and understanding how to manipulate equations is key to solving algebra problems effectively.

• Identify the given equations and what needs to be solved.
• Apply appropriate algebraic steps to simplify the equations.
• Ensure each step follows logically from the one before it.
• Check your solution by substituting back into the original equations.

By continually practicing these skills, solving algebra problems becomes much more intuitive and manageable.

Step-by-Step Solutions

The beauty of breaking down a problem into step-by-step solutions lies in its clarity and digestibility. By handling one step at a time, complex problems become much more manageable. In this context, being able to follow each step ensures that you can trace the logic and process behind finding the solution.

Let's consider the process used in our exercise. Starting with one equation solved for one variable simplifies the entire process. After this simplification, the rest of the equations are solved step by step, making sure each stage is understood before moving to the next one.

• Simplify: Simplify equations wherever possible to make calculations easier.
• Substitute: Substitute known values step-by-step to solve for unknown variables.
• Validate: Check final answers to ensure they satisfy all original equations.

Following these detailed procedures not only aids in solving the current problem but also builds a framework for approaching future problems with confidence.