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Chapter 6: Problem 51

Solve. $$\frac{4-x}{13}=\frac{11}{26}$$

Short Answer

Expert verified
The solution to the equation is \(x = -1.5\)

Step by step solution

01

- Multiply through by the denominator

Start by getting rid of the fractions. To do this, identify the common multiple of 13 and 26 which is 26. Multiply through by 26 to clear the fractions:\[26 \cdot \frac{4-x}{13} = 26 \cdot \frac{11}{26}\]This simplifies to:\[2(4-x)=11\]
02

- Distribute and simplify

Next, distribute the 2 in the left side of the equation and simplify:\[8-2x=11\]
03

- Solve for 'x'

Lastly, solve for 'x'. To do so, subtract 8 from both sides and divide by -2:\[-2x = 11 - 8\]\[-2x = 3\]\[x = \frac{3}{-2}\]

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Key Concepts

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

Equation Solving
Solving an equation is like cracking a puzzle where the answer tells you the value of a variable that makes the equation true. Here's how we can approach solving:
  • Identify the equation and determine what operations are affecting the variable.
  • Undo these operations step by step to isolate the variable.
  • Always perform the same operation on both sides to keep the equation balanced.
By applying these principles, we ensure that we are working systematically towards finding the solution without altering the foundational truth of the equation. With practice, solving equations becomes a straightforward exercise in reverse operations.
Fractions
Fractions can be tricky because they represent parts of a whole. They often appear in equations and can seem overwhelming at first. Here's the breakdown:
  • A fraction is expressed as \( \frac{a}{b} \), where \( a \) is the numerator and \( b \) is the denominator.
  • The denominator tells you into how many equal parts the whole is divided.
  • When solving equations, it is often useful to eliminate fractions by multiplying through by the common denominator.
In general, dealing with fractions becomes much easier once you become more familiar with their properties and how they operate within equations.
Linear Equations
Linear equations are fundamental in algebra, representing equations of a straight line. Here's what makes them special:
  • A linear equation is typically in the form \( ax + b = c \).
  • The highest power of the variable is one, which guarantees a straight line graph if plotted.
  • Solving a linear equation involves operations to isolate the variable on one side of the equation.
Linear equations are straightforward once you get the hang of solving them because they don't involve any exponents or complicated functions. They're continuous and predictable, making them a great starting point in algebraic learning.

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