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Chapter 5: Problem 15

Solve the equations. $$ 4 t+2(t+1)-5 t=13 $$

Short Answer

Expert verified
Question: Solve the equation: \(4 t+2(t+1)-5 t=13\) Answer: \(t = 11\)

Step by step solution

01

Distribute the coefficient inside the parentheses

First, we distribute the coefficient '2' inside the parentheses: $$ 4 t+2(t+1)-5 t=13 $$ $$ 4t + 2t + 2 - 5t = 13 $$
02

Combine like terms on both sides of the equation

Now, combine the like terms (all the terms containing 't'): $$ (4t - 5t + 2t) + 2 = 13 $$ $$ t + 2 = 13 $$
03

Isolate the variable 't'

In this step, we want to isolate 't' on one side of the equation. To do this, subtract '2' from both sides of the equation: $$ t + 2 - 2 = 13 - 2 $$ $$ t = 11 $$ So, the solution to the equation is \(t = 11\).

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Key Concepts

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

Distributive Property
The distributive property is a very useful tool in algebra that allows us to simplify expressions by multiplying a single term across terms inside parentheses. In our original exercise, we see it applied in the equation:
\[4t + 2(t+1) - 5t = 13\]
The term \(2(t+1)\) shows us a common case where the distributive property helps. Here's how we can apply it:
  • Multiply the coefficient \(2\) with each term inside the parentheses \((t+1)\).
  • This results in \(2 \cdot t + 2 \cdot 1\), which simplifies to \(2t + 2\).
Now, the equation is reduced to:
\[4t + 2t + 2 - 5t = 13\]
By doing this, we've simplified the expression and prepared it for the next steps. Using the distributive property is essential when dealing with equations that include parentheses, as it helps to break down and work through the equation systematically.
Combining Like Terms
Once the distributive property has been applied, the next step is to combine like terms. This step simplifies the equation further making it easier to solve. In the equation:
\[4t + 2t + 2 - 5t = 13\]
We can identify the 'like terms,' which are all the terms with the variable \(t\):
  • \(4t\)
  • \(2t\)
  • \(-5t\)
By combining these terms, we effectively gather all \(t\) terms together. This process is straightforward:
  • Start by summing \(4t\) and \(2t\) to get \(6t\).
  • Next, subtract \(5t\) from \(6t\) to arrive at \(t\).
After combining, we have:
\[t + 2 = 13\]
This combination process is crucial as it makes the equation much simpler, revealing the structure needed for further solutions.
Isolation of Variable
The final step in solving the equation is isolating the variable, which involves getting the variable by itself on one side of the equation. After combining like terms, our equation is:
\[t + 2 = 13\]
To isolate \(t\), we need to remove the constant term (\(+2\)) from the left side. This is achieved by performing the same arithmetic operation on both sides of the equation:
  • Subtract \(2\) from both sides of the equation.
  • This operation looks like: \[t + 2 - 2 = 13 - 2\].
Doing this simplifies the equation to:
\[t = 11\]
Now, the variable \(t\) is isolated, revealing the solution. Isolating the variable is one of the most important steps in solving equations, as it underscores the transformation of the equation into its simplest possible form.

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Most popular questions from this chapter

Which line has the greater (a) Slope? (b) \(y\) -intercept? $$ y+2=3(x-1), \quad y=6-50 x $$

Solve the system of equations graphically. $$ \left\\{\begin{array}{r} 2 x+5 y=7 \\ -3 x+2 y=1 \end{array}\right. $$

For the system $$ \left\\{\begin{array}{l} 2 x+3 y=5 \\ 4 x+6 y=n \end{array}\right. $$ what must be true about \(n\) in order for there to be many solutions?

Put the equation \(y=3 x t+2 x t^{2}+5\) in the form \(y=b+m x .\) What are the values of \(b\) and \(m ?\) [Note: Your answers could include \(t\).]

Are the lines parallel? $$ y=7+4(x-2) ; y=8+2(2 x+3) $$
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