Problem 31 - Elementary Algebra Within Reach

Question

In your own words, explain the basic steps in solving a system of linear equations by the method of substitution.

Step-by-Step Solution

Answer

The substitution method involves: 1) solving one equation for one variable, 2) substituting this expression into the other equation, 3) solving the resulting single-variable equation, 4) substituting this value back into one of the original equations to solve for the other variable, and 5) verifying the solution by substitifound into the original equations

Step 1: Understanding the Concept

The method of substitution involves solving one of the equations for one variable in terms of the other variable. This provides an expression that can be substituted into the other equation.

Step 2: Solving one of the Equations

Choose one of the equations to solve for a specific variable. The choice of which equation and which variable to solve for depends on which will simplify the problem most effectively.

Step 3: Substituting into the Other Equation

Substitute the expression obtained in the previous step into the other equation. This results in an equation with only one variable.

Step 4: Solving the Single Variable Equation

Solve this resulting single-variable equation. This will provide the value for one of the variables.

Step 5: Finding the Second Variable

Substitute the solved value into any of the original equations and solve for the other variable. This will provide the value for the second variable.

Step 6: Verification by Substitution

Finally, substitute the obtained values of the variables into both the original equations to verify that they satisfy the system of equations

Key Concepts

Method of Substitution

The method of substitution is a powerful tool used for solving systems of linear equations. It is particularly useful when the equations involve two variables. Here's a simple breakdown of this method:

• First, you choose one of the equations and solve it for one variable in terms of the other. This is typically done by isolating the variable on one side of the equation.
• After doing so, you'll get an expression that you can plug into the other equation. This substitution reduces the number of unknowns in the equation from two to one, simplifying your task significantly.

By carefully following these steps, you can discover solutions efficiently. The method of substitution is all about simplifying complex problems to manageable tasks by making strategic choices at each step.

Solving Equations

When you have performed the substitution and have a new equation with one variable, it's time to solve this equation. The procedure involves the following:

• Perform any necessary algebraic operations to isolate the variable on one side of the equation.
• Use techniques such as adding, subtracting, multiplying, or dividing both sides until you isolate the variable completely.

Keep your calculations clear and systematic. This will help avoid mistakes and lead you to find the correct value for the variable. Once you have this value, you can proceed to find the other variable using one of the original equations.

Algebraic Expressions

Algebraic expressions are key in the method of substitution. They allow us to express a variable in terms of another:

• An algebraic expression typically involves variables, constants, and arithmetic operations such as addition or subtraction.
• When you rearrange an equation during substitution, you're crafting a new algebraic expression.

Think of algebraic expressions as adaptable building blocks. They form the bridge that connects one equation to another, helping you to solve the system bit by bit.

Verification of Solutions

Once you have found values for the variables, it's crucial to verify these solutions. This step ensures accuracy:

• Substitute the values back into both original equations and check if they satisfy those equations.
• If both equations hold true with your found values, then you have correctly solved the system.

Verification acts as a safeguard against errors. It’s a straightforward yet essential task to confirm that your solutions truly work within the context of the problem.