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Chapter 12: Problem 1

Solve the equation: \(3 x+4=8-x\).

Short Answer

Expert verified
x = 1

Step by step solution

01

Isolate the variable on one side

Add x to both sides of the equation to get all the variable terms on one side:\[3x + 4 + x = 8 - x + x\]Which simplifies to:\[4x + 4 = 8\]
02

Eliminate the constant term from the variable side

Subtract 4 from both sides to isolate the variable term:\[4x + 4 - 4 = 8 - 4\]Which simplifies to:\[4x = 4\]
03

Solve for x

Divide both sides by 4 to solve for x:\[\frac{4x}{4} = \frac{4}{4}\]Which simplifies to:\[x = 1\]

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Key Concepts

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

Isolate the Variable
To solve linear equations effectively, the first step is to isolate the variable on one side of the equation. This essentially means we want all terms containing the unknown variable (in this case, x) to be on one side, and all constants on the other.

In our example:
  • Start with the equation: 3x + 4 = 8 - x.
  • Add x to both sides to move the variable term from the right-hand side to the left-hand side. By doing this, we get 3x + x + 4 = 8. It simplifies down to 4x + 4 = 8.
So, by adding x on both sides, we grouped all x terms together on the left side. This is our step towards isolating the variable.

Remember, whatever operation you perform on one side of the equation, you must do the exact same operation to the other side. This keeps the equation balanced.
Eliminate the Constant Term
The next key step is to eliminate the constant term from the side containing the variable. This helps in further isolating the variable.

Continuing with our example:
  • We have the equation 4x + 4 = 8.
  • Subtract 4 from both sides to get rid of the constant term on the left-hand side where our variable is. The operation looks like 4x + 4 - 4 = 8 - 4, which simplifies to 4x = 4.
This step is crucial because it ensures the variable term stands alone, making it clear and straightforward to solve for the variable in the next steps.

Just like before, whatever operation you perform on one side, do the same on the other side. This maintains the balance in the equation.
Divide to Solve for the Variable
Finally, we solve for the variable by dividing both sides of the equation by the coefficient of the variable.

In the equation we derived:
  • We have 4x = 4.
  • To solve for x, divide both sides by 4, resulting in x = 4/4, which simplifies to x = 1.
This method of dividing both sides by the coefficient ensures that the variable stands alone and provides the value directly.

In summary:
  • We started with 3x + 4 = 8 - x.
  • Reorganized it to 4x = 4 by isolating the variable and eliminating the constant term.
  • Divided both sides by 4 to solve for x, giving us x = 1.
By understanding and applying these steps, isolating the variable, eliminating the constant term, and then dividing to solve for the variable, you can confidently solve linear equations.

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Most popular questions from this chapter

Presale Order A wireless store owner takes presale orders for a new smartphone and tablet. He gets 340 preorders for the smartphone and 250 preorders for the tablet. The combined value of the preorders is \(\$ 486,000 .\) If the price of a smartphone and tablet together is \(\$ 1665,\) how much does each device cost?

Determine whether the product is defined. If it is defined, find the product; if it is not write "not defined." $$ \left[\begin{array}{rrr} 4 & -2 & 3 \\ 0 & 1 & 2 \\ -1 & 0 & 1 \end{array}\right]\left[\begin{array}{rr} 2 & 6 \\ 1 & -1 \\ 0 & 2 \end{array}\right] $$

Solve each system of equations. If the system has no solution, state that it is inconsistent. $$ \left\\{\begin{array}{r} x+2 y-z=-3 \\ 2 x-4 y+z=-7 \\ -2 x+2 y-3 z=4 \end{array}\right. $$

Painting a House Three painters (Beth, Dan, and Edie), working together, can paint the exterior of a home in 10 hours (h). Dan and Edie together have painted a similar house in \(15 \mathrm{~h}\). One day, all three worked on this same kind of house for \(4 \mathrm{~h},\) after which Edie left. Beth and Dan required 8 more hours to finish. Assuming no gain or loss in efficiency, how long should it take each person to complete such a job alone?

Based on material learned earlier in the course. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for the final exam. Graph: \(f(x)=\frac{2 x^{2}-x-1}{x^{2}+2 x+1}\)
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