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Chapter 2: Problem 30

Solve the equation (if possible). $$\frac{10 x+3}{5 x+6}=\frac{1}{2}$$

Short Answer

Expert verified
The solution to the equation is \(x = 0\).

Step by step solution

01

Eliminate the Fraction

Multiply both sides of the equation by (5x + 6) to get rid of the fraction, giving \(10x + 3 = \frac{1}{2} \cdot (5x + 6)\).
02

Simplify the Equation

Expand the right side of the equation, \(10x + 3 = 2.5x + 3\), then subtract 2.5x from both sides to get \(7.5x + 3 = 3\).
03

Solve for x

Subtract 3 from both sides to get \(7.5x = 0\), then divide both sides by 7.5 to get \(x = 0\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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鈥檚 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.
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.

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Most popular questions from this chapter

Given that the solutions of a quadratic equation are \(x=(-b \pm \sqrt{b^{2}-4 a c}) /(2 a),\) show that the sum of the solutions is \(S=-b / a\).

Solving an Equation Involving Fractions Find all solutions of the equation. Check your solutions. $$8\left(\frac{t}{t-1}\right)^{2}-2\left(\frac{t}{t-1}\right)-3=0$$

Two planes leave simultaneously from Chicago's O'Hare Airport, one flying due north and the other due east (see figure). The northbound plane is flying 50 miles per hour faster than the eastbound plane. After 3 hours, the planes are 2440 miles apart. Find the speed of each plane.

Graphical Analysis (a) use a graphing utility to graph the equation, (b) use the graph to approximate any \(x\) -intercepts of the graph, (c) set \(y=0\) and solve the resulting equation, and (d) compare the result of part (c) with the \(x\) -intercepts of the graph. $$y=3 x-3 \sqrt{x}-4$$

Solving an Equation Involving an Absolute Value Find all solutions of the equation algebraically. Check your solutions. $$|x+1|=x^{2}-5$$
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