Solving the Equation 4(5-x) = 8 – Unraveling the Unknown

Equations, such as 4(5-x) = 8, hold a captivating allure in mathematics as they represent the delicate balance between known and unknown quantities. Solving equations is a powerful tool to unveil the hidden values within. Among the myriad equations that perplex and engage mathematicians, the equation 4(5-x) = 8 stands as an intriguing puzzle waiting to be solved.

In this article, we embark on a journey into the depths of this equation, unraveling the methods and techniques required to find the elusive value of x that satisfies the equation. By exploring the intricate steps of equation-solving. Join us as we navigate the twists and turns of solving the equation 4(5-x) = 8, and witness the beauty of mathematical problem-solving unfold before your eyes.

Defining 4(5-x) = 8

Solving the equation 4(5-x) = 8 involves finding the value(s) of the variable x that make the equation true. The equation represents an algebraic relationship between the unknown variable x and known quantities, in this case, the numbers 4, 5, and 8.

By performing a series of algebraic manipulations and simplifications, the goal is to determine the specific value(s) of x that satisfy the equation. Solving equations is a fundamental process in mathematics and serves as a vital tool in various applications, allowing us to discover unknown quantities and make mathematical connections.

By unraveling the mystery of equation 4(5-x) = 8, we unlock a gateway to the world of problem-solving and unveil the solutions that bring clarity to the mathematical realm.

Solving the Equation 4(5-x) = 8

To solve the equation 4(5-x) = 8, follow these steps:

Step 1

Distribute the 4 on the left side of the equation: Multiply 4 by each term inside the parentheses.

4 * 5 – 4 * x = 8

20 – 4x = 8

Step 2

Simplify the equation: Combine like terms by subtracting 20 from both sides.

20 – 20 – 4x = 8 – 20

-4x = -12

Step 3

Divide both sides by -4: To isolate the variable x, divide both sides of the equation by -4.

(-4x) / -4 = (-12) / -4

x = 3

Step 4

Verify the solution: To confirm that the solution x = 3 is correct, substitute it back into the original equation and check if both sides are equal.

4(5 – 3) = 8

4(2) = 8

8 = 8

Since both sides of the equation are equal when x = 3, we can conclude that the solution x = 3 satisfies the original equation.

Therefore, the solution to the equation 4(5-x) = 8 is x = 3.

Applications

Solving the equation 4(5-x) = 8 has practical applications in various fields. Here are some examples:

Physics and Engineering

Equations are vital in physics and engineering, where mathematical models describe physical phenomena. Solving equations helps determine unknown variables or quantities in various systems. In this context, solving equations like 4(5-x) = 8 allows engineers and physicists to find the values of variables involved in mathematical representations of physical processes. This can be useful for determining distances, velocities, forces, or other physical parameters.

Finance and Economics

Equations frequently arise in finance and economics to model various scenarios and relationships. Solving equations in these contexts can help with investments, interest rates, profit margins, or economic forecasting calculations. By solving equations like 4(5-x) = 8, financial analysts and economists can determine unknowns such as pricing, revenue, or market equilibrium.

Optimization and Planning

Equations are often used in optimization problems where the goal is to find the best possible solution. Researchers and planners can optimize resource allocation, scheduling, production, or logistics by solving equations. Equations like 4(5-x) = 8 can represent constraints or objectives that must be satisfied in an optimization problem.

Data Analysis and Modeling

Equations are fundamental in data analysis and modeling. Solving equations allows researchers to estimate unknown parameters or variables based on observed data. Analysts can make predictions, draw conclusions, or validate hypotheses by solving equations related to statistical models or regression analysis.

Computer Science and Programming

Equations play a role in computer science and programming, particularly in algorithms and numerical methods. Solving equations helps solve problems encountered in algorithm design, simulations, or optimization routines. Equations like 4(5-x) = 8 can be part of algorithms or mathematical calculations in various domains, including machine learning, computer graphics, or cryptography.

Everyday Life

Solving equations is not limited to academic or specialized fields. In everyday life, equations are encountered in various contexts, such as budgeting, home improvement, or personal finance. Solving equations helps individuals make informed decisions, solve practical problems, or plan their activities effectively.

By solving equations like 4(5-x) = 8, we apply fundamental problem-solving skills that find applications in numerous disciplines and situations. Whether in physics, finance, optimization, data analysis, computer science, or everyday life, solving equations is a valuable tool for decision-making, understanding relationships, and advancing our understanding of the world around us.

Exercise

Example 1

Solve the equation 4 * (5-x) + 3 = 11.

Solution

To solve this equation, we follow similar steps as before:

4 * (5-x) + 3 = 11

20 – 4x + 3 = 11

-4x + 23 = 11

-4x = 11 – 23

-4x = -12

x = (-12) / (-4)

x = 3

So, the solution to the equation 4 * (5-x) + 3 = 11 is x = 3.

Example 2

Find the value of x that satisfies the equation 4 *  (5-x) = 16.

Figure-2.

Solution

We can solve this equation as follows:

4 *  (5-x) = 16

20 – 4x = 16

-4x = 16 – 20

-4x = -4

x = (-4) / (-4)

x = 1

Therefore, the value of x that satisfies the equation 4 *  (5-x) = 16 is x = 1.

Example 3

Determine the solution set for the equation 4 *  (5-x) = -8.

Figure-3.

Solution

To find the solution set, we proceed as follows:

4 * (5-x) = -8

20 – 4x = -8

-4x = -8 – 20

-4x = -28

x = (-28) / (-4)

x = 7

Hence, the solution set for the equation 4 * (5-x) = -8 is x = 7.

Example 4

Solve the equation 4 *  (5-x) = 0.

Solution

To solve this equation, we proceed as follows:

4 * (5-x) = 0

20 – 4x = 0

-4x = -20

x = (-20) / (-4)

x = 5

So, the solution to the equation 4 * (5-x) = 0 is x = 5.

Example 5

Solve the equation 2 * (4-x) – 1 = 20.

Solution

To solve the equation 2 * (4-x) – 1 = 20, we can follow these steps:

Distribute the 2 on the left side of the equation:

2 * (4-x) – 1 = 20

8 – 2x – 1 = 20

Combine the constants on the left side of the equation:

7 – 2x = 20

To isolate the variable term, subtract 7 from both sides of the equation:

7 – 7 – 2x = 20 – 7

-2x = 13

To solve for x, divide both sides of the equation by -2:

(-2x) / -2 = 13 / -2

x = -6.5

So, the solution to the equation  2 * (4-x) – 1 = 20 is x = -6.5.

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