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Chapter 2: Problem 44

Solve each equation, and check your solution. $$ 1.9 t-6=0.9 t $$

Short Answer

Expert verified
The solution is \(t = 6\). Substituting \(t = 6\) confirms both sides of the equation are equal.

Step by step solution

01

Move all terms involving 't' to one side

Subtract 0.9t from both sides of the equation to get all the 't' terms on one side: \(1.9t - 0.9t - 6 = 0\)
02

Simplify the equation

Combine like terms on the left side: \(1.0t - 6 = 0\)
03

Isolate the variable 't'

Add 6 to both sides of the equation to isolate 't': \(t = 6\)
04

Check the solution

Substitute \(t = 6\) back into the original equation to ensure it is correct: \(1.9(6) - 6 = 0.9(6)\) Calculate both sides: \(11.4 - 6 = 5.4\) Both sides equal \(5.4\), thus proving the solution \(t = 6\) is correct.

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Key Concepts

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

Understanding Linear Equations
Linear equations are algebraic expressions where each term is either a constant or the product of a constant and a single variable. They are essential in mathematics because they form the basis for more complicated equations.
Linear equations typically appear in the form: ax + b = c.

Here, 'a' and 'b' are constants, 'x' is the variable we're trying to solve for, and 'c' is the constant on the other side of the equality.
Solving linear equations involves finding the value of the variable that makes the equation true.
The solution must satisfy both sides of the equation when the variables are substituted back.
Isolating Variables
Isolating the variable means rearranging an equation so that the variable we need to solve for stands by itself on one side of the equation.
To isolate the variable 't' in our problem: 1.9t - 6 = 0.9t, we need to follow a few steps:
  • First, move all terms involving 't' to one side. Subtract 0.9t from both sides, giving us: 1.9t - 0.9t - 6 = 0.
  • Then handle the constants so that the term with the variable can be simplified. This simplification occurs by combining like terms.
When a variable is isolated, it is easier to find its value by performing operations that leave the variable by itself.
Combining Like Terms
Combining like terms is essential when solving linear equations as it simplifies the expression, making it easier to isolate variables.
Like terms are terms that contain the same variable raised to the same power.
In our problem, 1.9t - 0.9t - 6 = 0, the terms 1.9t and 0.9t are like terms because both contain 't'.
To combine them:
  • Subtract 0.9t from 1.9t resulting in 1.0t.
This makes the equation simpler: 1.0t - 6 = 0.
Combining like terms reduces the complexity and helps in solving the equation more easily.
Checking Solutions
Checking the solution of a linear equation ensures that the solution is correct.
After isolating the variable and finding its value, substitute it back into the original equation to see if both sides are equal.
In our example, we found t = 6. To check:
Substitute t = 6 back into the original equation: 1.9(6) - 6 = 0.9(6).
Calculate both sides:
1.9(6) = 11.4 and 0.9(6) = 5.4.
So, 11.4 - 6 = 5.4.
Since both sides are equal (5.4 = 5.4), the solution t = 6 is verified as correct. This process is crucial to ensure no mistakes were made during the calculation.

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