Problem 20 - Fundamentals of Mathematics

Question

Verify that each given value is a solution to the given equation. $$4 x-5=6 x-20, x=\frac{15}{2}$$

Step-by-Step Solution

Answer

Yes, \(x = \frac{15}{2}\) is a solution to the equation.

Step 1: Understand the Problem

We need to verify if the given value \(x = \frac{15}{2}\) satisfies the equation \(4x - 5 = 6x - 20\). This means we will substitute \(\frac{15}{2}\) into the equation in place of \(x\) and check if both sides of the equation are equal.

Step 2: Substitute the Value of x

Replace \(x\) in the equation with \(\frac{15}{2}\):\[4\left(\frac{15}{2}\right) - 5 = 6\left(\frac{15}{2}\right) - 20\]

Step 3: Calculation of Left Side

Calculate the left side of the equation:\[4 \times \frac{15}{2} = \frac{60}{2} = 30\]Now subtract 5:\[30 - 5 = 25\]

Step 4: Calculation of Right Side

Calculate the right side of the equation:\[6 \times \frac{15}{2} = \frac{90}{2} = 45\]Now subtract 20:\[45 - 20 = 25\]

Step 5: Compare Both Sides

Both sides of the equation are equal (25 = 25) after substitution, which means \(x = \frac{15}{2}\) is a solution to the equation.

Key Concepts

Algebra

Algebra is a fascinating branch of mathematics that deals with symbols and the rules for manipulating these symbols. These symbols often represent numbers, and the symbols and variables make it possible to formulate and solve equations. Think of algebra as a language. Instead of words, it uses letters and symbols to solve everyday problems.

In the given problem, we are working with a linear equation, which is any equation that models a straight line when plotted on a graph. The equation given is linear because it only involves the variables raised to the first power. This type of equation is common in algebra.

• Linear equations appear in the form of ax + b = c.
• They are called linear because they form straight lines when graphed.

Understanding algebra is crucial because it forms the basis for more complex mathematics like calculus and geometry. It allows you to change and structure mathematical models that help solve real-world problems.

Substitution Method

The substitution method is a simple technique used to verify solutions to equations. It's a two-step process involving replacing a variable with a given value to check if it satisfies the equation.

In the original exercise, we're tasked with verifying whether the value \(x = \frac{15}{2}\) is a solution for the equation \(4x - 5 = 6x - 20\). Substitution step-by-step involves:

• Replacing variable \(x\) with \(\frac{15}{2}\) in the equation.
• Calculating both sides of the equation to ensure they are equal.

This method is very effective because it directly confirms the accuracy of a solution by plugging it back into the original equation. If both sides come out the same, the solution is verified. This process is particularly useful in algebra when we have equations with a clear variable being solved for.

Solving Linear Equations

Solving linear equations is all about finding values of variables that make the equation true. In a linear equation, you will often see terms involving variables that can be added, subtracted, multiplied, or divided. These operations follow specific rules to isolate the variable and find its value.

For the given problem, you first substitute the known value into the equation, and then perform arithmetic operations to simplify both sides.

• Calculate the arithmetic operations on the left and right sides separately.
• Ensure both sides equal each other after substitution.

In this case, substituting \(x = \frac{15}{2}\) in \(4x - 5 = 6x - 20\) gives both sides a value of 25, verifying that the solution is correct. This step is crucial in algebra, as it shows that you understand both the concept of the equations and the method of substitution.
By practicing solving linear equations with the substitution method, you build significant mathematical skills that are necessary for solving more complex problems in mathematics and other scientific fields.