Problem 108 - Introductory Algebra for College Students

Question

Determine whether the given number is a solution of the equation. $$\frac{6 m-5}{11}=\frac{3 m-2}{5} ;-1$$

Step-by-Step Solution

Answer

-1 is the solution of the given equation.

Step 1: Substitution

Substitute the value of 'm' in the equation with the given number -1. The equation turns into \(\frac{6(-1)-5}{11}=\frac{3(-1)-2}{5}\).

Step 2: Simplification

Simplify the left and right side of the equation respectively to get \(\frac{-6-5}{11}=\frac{-3-2}{5}\). It then becomes \(-1=-1\). As both sides are equal, -1 is the solution to the equation.

Key Concepts

Substitution Method

The substitution method is an essential technique in algebra, particularly when it comes to evaluating whether a given number is a solution to an equation. Imagine an equation as a balance scale; both sides should weigh the same for it to balance perfectly. In our case, we are given an equation and a number. To use the substitution method, take the given number and replace the variable with it throughout the equation.

For instance, if we have the variable 'm' and we're given the number -1, we substitute 'm' with -1. This involves replacing every instance of 'm' in the equation with -1. Substitution allows us to transform the algebraic equation into a numerical one, which can then be evaluated for equality. If both sides equal the same value after the substitution, then the given number is indeed a solution. This method is straightforward and especially practical for determining solutions to equations quickly.

Simplifying Expressions

When dealing with algebraic equations, simplifying expressions is a critical step that makes it easier to understand and solve them. Simplifying an expression involves combining like terms and performing any arithmetic operations to reduce the expression to its simplest form.

For example, after substituting a value into an equation, you might end up with an expression like \(\frac{6(-1)-5}{11}\). To simplify this, multiply 6 by -1, subtract 5, and divide by 11, to ultimately arrive at a single numerical value. Simplification might require you to work with fractions, distribute multiplication over addition, or combine terms. It's like tidying up your workspace so that you can clearly see the task at hand. In the context of solving equations, a simplified expression on each side of the equals sign paves the way for easier comparison.

Equation Solutions

The concept of equation solutions is foundational in algebra. A solution to an equation is a value that, when substituted for the variable, makes the equation a true statement. In other words, it's the number that balances our metaphorical scale.

After simplifying both sides of the equation, if you find that they hold the same value, then your substitution reveals a true solution. In our exercise, when we get \( -1 = -1 \), we confirm that the equation balances out, affirming -1 as the solution. It's like solving a puzzle—finding the right piece that fits perfectly. To master solving equations, it's crucial to practice various methods, understand the principles of equality and operations, and always check your solutions.