Problem 10
Question
Solve each of the equations. $$x=5.3-0.06 x$$
Step-by-Step Solution
Verified Answer
x = 5.0
1Step 1: Identify Like Terms
The equation is given as \( x = 5.3 - 0.06x \). Our first task is to group the like terms together. In this case, this means moving the term \(-0.06x\) to the left side of the equation to combine it with \(x\). This involves adding \(0.06x\) to both sides of the equation, giving us \(x + 0.06x = 5.3\).
2Step 2: Simplify the Equation
Now, simplify the left side of the equation by combining the \(x\) terms. The expression \(x + 0.06x\) can be written as \(1x + 0.06x\), which simplifies to \(1.06x\). This gives us the equation \(1.06x = 5.3\).
3Step 3: Solve for x
To isolate \(x\), divide both sides of the equation by \(1.06\). This results in \(x = \frac{5.3}{1.06}\).
4Step 4: Calculate the Solution
Finally, perform the division \(\frac{5.3}{1.06} \) to find the value of \(x\). This gives us \(x \approx 5.0\).
Key Concepts
Variable IsolationSimplifying EquationsCombining Like Terms
Variable Isolation
When solving linear equations, one of the critical steps is variable isolation. This means getting the variable you're trying to solve for, often denoted as \(x\), all by itself on one side of the equation.
Consider the equation given: \(x = 5.3 - 0.06x\). Here, \(x\) appears on both sides of the equation. The first goal is to gather all the \(x\) terms on one side. In this case, we add \(0.06x\) to both sides of the equation, thus ensuring we are correctly isolating the variable.
By bringing the \(x\) terms together on one side, we set ourselves up for easier simplification and management of the equation, crucial for finding the eventual value.
Consider the equation given: \(x = 5.3 - 0.06x\). Here, \(x\) appears on both sides of the equation. The first goal is to gather all the \(x\) terms on one side. In this case, we add \(0.06x\) to both sides of the equation, thus ensuring we are correctly isolating the variable.
By bringing the \(x\) terms together on one side, we set ourselves up for easier simplification and management of the equation, crucial for finding the eventual value.
Simplifying Equations
After bringing terms together for variable isolation, the next crucial step is simplifying the equation. Simplifying equations involves reducing the equation's complexity to get a clearer path towards the solution.
With our example, after isolating \(x\), we end up with \(x + 0.06x = 5.3\). Simplification in this context means combining these terms into a single, easier-to-manage expression. Here, we see \(x + 0.06x\) can be simplified to \(1.06x\), as you factor in the implicit '1' before \(x\).
Simplification reduces errors and makes calculations more straightforward, ensuring a systematic approach towards solving equations.
With our example, after isolating \(x\), we end up with \(x + 0.06x = 5.3\). Simplification in this context means combining these terms into a single, easier-to-manage expression. Here, we see \(x + 0.06x\) can be simplified to \(1.06x\), as you factor in the implicit '1' before \(x\).
Simplification reduces errors and makes calculations more straightforward, ensuring a systematic approach towards solving equations.
Combining Like Terms
Combining like terms is a fundamental part of solving equations and involves grouping similar terms together. This concept is essential when dealing with equations to make them simpler and easier to solve.
Like terms are terms that have the same variables raised to the same powers. In our equation, both \(x\) and \(-0.06x\) are like terms because they involve the variable \(x\).
In order to simplify, these like terms are combined by performing arithmetic operations on their coefficients. So, from \(x + 0.06x\), you simply add the coefficients, 1 and 0.06, resulting in \(1.06x\).
Combining like terms helps to streamline an equation, reducing it to its most simplified form and aligning it towards a clear path to the solution.
Like terms are terms that have the same variables raised to the same powers. In our equation, both \(x\) and \(-0.06x\) are like terms because they involve the variable \(x\).
In order to simplify, these like terms are combined by performing arithmetic operations on their coefficients. So, from \(x + 0.06x\), you simply add the coefficients, 1 and 0.06, resulting in \(1.06x\).
Combining like terms helps to streamline an equation, reducing it to its most simplified form and aligning it towards a clear path to the solution.
Other exercises in this chapter
Problem 10
For Problems \(1-12\), solve each of the equations. These equations are the types you will be using in Problems 13-40. $$ \frac{10}{3} r+\frac{10}{3}(r-3)=90 $$
View solution Problem 10
For Problems \(1-10\), solve for the specified variable using the given facts. (Objective 1) Solve \(A=P+P r t\) for \(P\) if \(A=1032, r=0.06\), and \(t=12\).
View solution Problem 10
Solve each of the equations. $$\frac{x-2}{6}=\frac{x-6}{8}$$
View solution Problem 11
For Problems 1-12, solve each equation. You will be using these types of equations in Problems \(13-41\). $$ 3 t=\frac{11}{2}\left(t-\frac{3}{2}\right) $$
View solution