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Chapter 1: Problem 53

Solve each of the given equations for \(\mathrm{x}\). $$19 x+35=10$$

Short Answer

Expert verified
The solution is \(x = -\frac{25}{19}\).

Step by step solution

01

Isolate the Variable Term

To solve the equation \(19x + 35 = 10\), we first need to isolate the term with the variable \(x\) on one side. To do this, subtract 35 from both sides of the equation:\[19x + 35 - 35 = 10 - 35\]This simplifies to:\[19x = -25\]
02

Solve for x

Now that we have isolated the term with \(x\), we need to solve for \(x\) itself. Divide both sides of the equation by 19 to find the value of \(x\):\[x = \frac{-25}{19}\]The solution to the equation is:\[x = -\frac{25}{19}\]

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Key Concepts

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

Isolating Variables
When we talk about isolating variables, it's about making the variable stand alone on one side of the equation. In math, this often means moving other numbers or terms away from the variable's side until it is by itself. This process helps us to clearly see what the variable equals.
To isolate a variable, you'll need to use basic arithmetic operations like addition, subtraction, multiplication, or division. This is done repeatedly until the variable is alone.
For instance, consider the equation from the original exercise:
  • We had the equation: \[19x + 35 = 10\]
  • Our goal was to get \(x\) by itself, so we subtracted 35 from both sides:\[19x + 35 - 35 = 10 - 35\]
  • The operation reduced to \[19x = -25\]
This leaves us with \(x\) isolated once we perform the division by 19, leading us to solve what \(x\) equals.
Solving Equations
Solving an equation involves finding the value of the variable that makes the equation true. Think of solving as a detective's job鈥攜ou're uncovering the mystery of what the variable represents.
Equations maintain balance, like a teeter-totter. Each side of the equation must be equal. When solving, we apply operations to both sides without disturbing this balance.
For example, in the exercise:
  • After isolating the variable to \[19x = -25\],
  • We needed to solve for \(x\). This involved dividing both sides by the coefficient of \(x\), which is 19:\[x = \frac{-25}{19}\]
Solving equations step by step ensures that every action we take makes sense and keeps the equation balanced, leading us to the correct solution.
Fractions in Algebra
Fractions can appear intimidating, but they are simply another form of numbers. They often show up in algebra when dividing values, especially during the last steps of solving equations.
Understanding how to work with fractions is crucial since they provide more precise answers that can be easier to interpret in some cases.
In the context of the given equation:
  • When we reached \[x = \frac{-25}{19}\],
  • it's essential to understand that this fraction indicates the exact division result of -25 by 19, rather than an approximation.
  • Instead of converting fractions to decimals immediately, it's often best to leave them in fraction form unless a decimal is needed for further calculations.
Fractions in algebra help us see the exact solution to an expression's division, making them an invaluable tool in more complicated equations.

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