Problem 45
Question
Solve the equation. Round the result to two decimal places. $$6.35 x-9.94=3.88+40.34 x$$
Step-by-Step Solution
Verified Answer
The solution to the equation is \( x = 1.84 \).
1Step 1: Group like terms
Combine the constant terms on one side of the equation, and x terms on the other. This gives: \[7.51x = 13.82\]
2Step 2: Solve for x
Next, divide each side of the equation by the coefficient of x (7.51) in order to solve for x. This gives: \[x = \frac{13.82}{7.51}\]
3Step 3: Compute the Value of x and Round Off
Perform the division to find the value of x and round off to two decimal places. This gives: \[x = 1.84\]
Key Concepts
Algebraic EquationsRounding NumbersMathematical Operations
Algebraic Equations
Algebraic equations are mathematical statements that show the equality between two expressions. They often involve one or more variables, such as \(x\), that you need to solve for. In the given exercise, for instance, the equation is \(6.35x - 9.94 = 3.88 + 40.34x\). Here, \(x\) stands as the variable that we're solving for.
The process begins with combining like terms, which means gathering all terms containing the variable \(x\) on one side of the equation and constant numbers on the other. This helps in simplifying the equation and making it easier to isolate \(x\). For example, in the provided problem, the mathematical strategy is to keep all \(x\) terms on one side. This activity transforms our initial equation into \(7.51x = 13.82\).
Learning to solve algebraic equations is a foundational skill in algebra, which in turn is fundamental for various other math topics such as calculus and statistics. Once you've isolated the variable terms, you can solve for the variable by performing the necessary mathematical operations.
The process begins with combining like terms, which means gathering all terms containing the variable \(x\) on one side of the equation and constant numbers on the other. This helps in simplifying the equation and making it easier to isolate \(x\). For example, in the provided problem, the mathematical strategy is to keep all \(x\) terms on one side. This activity transforms our initial equation into \(7.51x = 13.82\).
Learning to solve algebraic equations is a foundational skill in algebra, which in turn is fundamental for various other math topics such as calculus and statistics. Once you've isolated the variable terms, you can solve for the variable by performing the necessary mathematical operations.
Rounding Numbers
Rounding numbers is a useful technique in mathematics that simplifies numbers to a desired level of accuracy. It is especially crucial when the result of a calculation is a long decimal or when only an approximate value is needed.
In the context of solving equations, once you have isolated the variable and computed its actual value, you might need to round it for precision, as seen in this exercise. Here, after dividing 13.82 by 7.51, the exact value might be a longer decimal. Rounding it to two decimal places gives us \(x = 1.84\).
To round to two decimal places:
In the context of solving equations, once you have isolated the variable and computed its actual value, you might need to round it for precision, as seen in this exercise. Here, after dividing 13.82 by 7.51, the exact value might be a longer decimal. Rounding it to two decimal places gives us \(x = 1.84\).
To round to two decimal places:
- Look at the third decimal place. If it is 5 or more, increase the second decimal place by one.
- If it is less than 5, keep the second decimal place the same.
Mathematical Operations
Mathematical operations such as addition, subtraction, multiplication, and division are the foundational tools used to solve algebraic equations. In the provided exercise, these operations are crucial at every step.
First, when combining like terms, subtraction is applied to move terms between the equation's two sides—turning the situation into one variable term and one constant term. In our problem, you perform the operation \(6.35x - 40.34x = 3.88 + 9.94\) to simplify, then compute directly to ultimately isolate \(x\), leading you to \(7.51x = 13.82\).
Division is the next operation when finding the value of \(x\). Here, you divide the result of the constant term by the coefficient of \(x\). The operation is expressed as \(x = \frac{13.82}{7.51}\), ultimately giving us our answer post-rounding.
Understanding and correctly applying these mathematical operations allows you to not only solve equations but also to tackle more complex mathematical problems.
First, when combining like terms, subtraction is applied to move terms between the equation's two sides—turning the situation into one variable term and one constant term. In our problem, you perform the operation \(6.35x - 40.34x = 3.88 + 9.94\) to simplify, then compute directly to ultimately isolate \(x\), leading you to \(7.51x = 13.82\).
Division is the next operation when finding the value of \(x\). Here, you divide the result of the constant term by the coefficient of \(x\). The operation is expressed as \(x = \frac{13.82}{7.51}\), ultimately giving us our answer post-rounding.
Understanding and correctly applying these mathematical operations allows you to not only solve equations but also to tackle more complex mathematical problems.
Other exercises in this chapter
Problem 44
Complete these steps for the function. a. Tell whether the graph of the function opens up or down. b. Find the coordinates of the vertex. c. Write an equation o
View solution Problem 45
Use a calculator to evaluate the expression. Round the results to the nearest hundredth. $$\frac{3 \pm 4 \sqrt{6}}{3}$$
View solution Problem 45
Write the quadratic equation in standard form. Solve using the quadratic formula. $$2-3 x+x^{2}=0$$
View solution Problem 45
Use a graphing calculator to approximate the solution of the equation. $$ x^{2}-3 x-4=0 $$
View solution