The linear equation d – 10 – 2d + 7 = 8 + d – 10 – 3d may look intimidating at first glance, but it is actually quite simple to solve. This equation is an example of a linear equation, which is an equation in which the highest power of the variable is one. Linear equations are often used to model real-world situations, such as calculating the cost of a phone plan or determining the distance traveled by a car.
To solve this equation, we must first simplify it by combining like terms. Once we have simplified the equation, we can isolate the variable on one side of the equation by performing the same operation to both sides of the equation. In this case, we can isolate the variable d by adding 3d to both sides of the equation and then adding 3 to both sides of the equation. The resulting solution is d = 1.
Understanding Linear Equations
What is a Linear Equation?
A linear equation is an equation that forms a straight line when graphed. It is a mathematical expression that relates two variables, typically represented by x and y, in a straight line. Linear equations are commonly used in mathematics, science, engineering, and economics to model and analyze real-world situations.
The general form of a linear equation is y = mx + b, where m is the slope of the line and b is the y-intercept. The slope represents the rate of change of y with respect to x, while the y-intercept represents the value of y when x is equal to zero.
Solving Linear Equations
To solve a linear equation, we need to isolate the variable on one side of the equation. The following steps can be used to solve a linear equation:
- Simplify both sides of the equation using the order of operations and combine all same-side like terms.
- Use the appropriate properties of equality to combine opposite-side like terms with the variable term on one side of the equation and the constant term on the other.
- Divide or multiply as needed to isolate the variable.
Let’s apply these steps to the given equation, d – 10 – 2d + 7 = 8 + d – 10 – 3d:
- Simplify both sides of the equation: -d – 3 = -2d – 2
- Combine opposite-side like terms: d = 1
- The solution to the linear equation is d = 1.
In summary, linear equations are important mathematical tools used to model and analyze real-world situations. To solve a linear equation, we need to isolate the variable on one side of the equation using the appropriate properties of equality.
Solving the Given Linear Equation
Simplifying the Equation
The given linear equation is d – 10 – 2d + 7 = 8 + d – 10 – 3d. To solve this equation, we need to simplify it first. We can simplify the equation by combining like terms on both sides.
d – 2d + d – 3d – 10 + 7 – 10 + 8 = 0
Simplifying further, we get:
-3d – 5 = 0
Isolating the Variable
Now, we need to isolate the variable on one side of the equation. To do this, we can add 5 to both sides of the equation.
-3d = 5
Dividing both sides by -3, we get:
d = -5/3
Checking the Solution
To check if our solution is correct, we can substitute the value of d into the original equation and see if it holds true.
d – 10 – 2d + 7 = 8 + d – 10 – 3d
Substituting d = -5/3, we get:
-5/3 – 10 – 2(-5/3) + 7 = 8 – 5/3 – 10 – 3(-5/3)
Simplifying, we get:
-15/3 – 3.33 + 7 = 8 + 5/3 – 10 + 5
Which simplifies to:
-5 – 3.33 + 7 = 3.33 – 2
Simplifying further, we get:
-1.33 = 1.33
Since the equation is not true, our solution is incorrect.
In conclusion, the given linear equation d – 10 – 2d + 7 = 8 + d – 10 – 3d has no solution.
Common Mistakes to Avoid
When solving a linear equation, it’s easy to make mistakes. Here are some common mistakes to avoid.
Misunderstanding the Order of Operations
One common mistake when solving linear equations is misunderstanding the order of operations. It’s important to remember that you should always perform operations in the following order: parentheses, exponents, multiplication and division (from left to right), and finally addition and subtraction (from left to right).
Forgetting to Combine Like Terms
Another common mistake is forgetting to combine like terms. Like terms are terms that have the same variable raised to the same power. For example, 2x and 5x are like terms, but 2x and 5x^2 are not. When solving an equation, it’s important to combine like terms before proceeding.
Mistakes in Distributing the Negative Sign
Distributing the negative sign is another area where mistakes can be made. When distributing the negative sign, it’s important to remember to change the sign of each term that is being distributed. For example, -3(x + 2) should be expanded to -3x – 6, not -3x + 2.
By avoiding these common mistakes, you can improve your chances of solving linear equations correctly. Remember to always double-check your work and take your time to avoid making careless errors.
Applications of Linear Equations
Linear equations have a wide range of applications in various fields, from science to business. They are used to model real-world situations and to predict outcomes based on given data. Here are a few examples of how linear equations are used in different areas.
Linear equations can be used to solve everyday problems. For example, if you are driving at a constant speed, you can use a linear equation to determine how long it will take you to reach your destination. Similarly, if you know the distance between two cities and the speed of a train, you can use a linear equation to calculate the time it will take for the train to reach its destination.
Linear equations are commonly used in business to predict sales, profits, and other financial metrics. For instance, a company can use a linear equation to forecast sales based on past data. It can also use a linear equation to determine the break-even point, which is the point at which the company’s revenue equals its expenses.
Linear equations are used extensively in science to model physical phenomena. For example, scientists use linear equations to describe the relationship between force and acceleration. They also use linear equations to model the behavior of gases, liquids, and other substances.
In conclusion, linear equations are a powerful tool for modeling real-world situations and predicting outcomes based on given data. They have applications in many fields, including science, business, and everyday life.