Problem 30
Question
Solve the equation (if possible). $$\frac{10 x+3}{5 x+6}=\frac{1}{2}$$
Step-by-Step Solution
Verified Answer
The solution to the equation is \(x = 0\).
1Step 1: Eliminate the Fraction
Multiply both sides of the equation by (5x + 6) to get rid of the fraction, giving \(10x + 3 = \frac{1}{2} \cdot (5x + 6)\).
2Step 2: Simplify the Equation
Expand the right side of the equation, \(10x + 3 = 2.5x + 3\), then subtract 2.5x from both sides to get \(7.5x + 3 = 3\).
3Step 3: Solve for x
Subtract 3 from both sides to get \(7.5x = 0\), then divide both sides by 7.5 to get \(x = 0\).
Key Concepts
Eliminate Fractions in EquationsEquation SimplificationLinear Equation Solving
Eliminate Fractions in Equations
When faced with equations involving fractions, the initial step is to eliminate the fractions to simplify the problem. This often involves finding a common denominator and then multiplying each term in the equation by this common denominator. In the case of \(\frac{10x+3}{5x+6}=\frac{1}{2}\), we multiply both sides by the denominator \(5x + 6\) to eliminate the fraction.
This crucial step transforms a rational equation into a more straightforward linear equation, paving the way for simpler arithmetic operations and, ultimately, the isolation of the variable. It’s essential for students to comfortably manage this approach as it streamlines complex problems into more manageable ones.
This crucial step transforms a rational equation into a more straightforward linear equation, paving the way for simpler arithmetic operations and, ultimately, the isolation of the variable. It’s essential for students to comfortably manage this approach as it streamlines complex problems into more manageable ones.
Equation Simplification
Once fractions are eliminated, the next step is to simplify the equation. Simplification may involve distributing multiplication over addition or subtraction, combining like terms, and cancelling out terms to reduce the equation to its simplest form. In our exercise, after eliminating fractions, we expand the right side to simplify the equation, leading to \(10x + 3 = 2.5x + 3\).
Then, we perform operations such as subtracting \(2.5x\) from both sides. Simplification minimizes the potential for errors in subsequent steps and brings us closer to finding the value of the variable. Remember to perform the same operations on both sides of the equation to maintain equality, which is a fundamental principle in equation simplification.
Then, we perform operations such as subtracting \(2.5x\) from both sides. Simplification minimizes the potential for errors in subsequent steps and brings us closer to finding the value of the variable. Remember to perform the same operations on both sides of the equation to maintain equality, which is a fundamental principle in equation simplification.
Linear Equation Solving
The ultimate goal in solving rational equations is to find the value of the unknown variable. Once our equation is simplified to \(7.5x + 3 = 3\), it resembles a basic linear equation. Solving linear equations involves isolating the variable on one side of the equation.
In our example, we subtract \(3\) from both sides to get \(7.5x = 0\) and then divide by \(7.5\) to isolate \(x\). It yields \(x = 0\), the solution to the equation. The process of linear equation solving relies on understanding the properties of equality and mastery in applying arithmetic operations to manipulate the equation methodically until the variable is by itself.
In our example, we subtract \(3\) from both sides to get \(7.5x = 0\) and then divide by \(7.5\) to isolate \(x\). It yields \(x = 0\), the solution to the equation. The process of linear equation solving relies on understanding the properties of equality and mastery in applying arithmetic operations to manipulate the equation methodically until the variable is by itself.
Other exercises in this chapter
Problem 30
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