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Chapter 2: Problem 73

Solve each equation. Check each solution. $$ 20=-3(2 x+1)+7 x $$

Short Answer

Expert verified
The solution is \(x = 23\).

Step by step solution

01

Distribute the multiplication

First, distribute the \(-3\) across the term \((2x + 1)\). This gives:\[20 = -3(2x) - 3(1) + 7x \]which simplifies to: \[20 = -6x - 3 + 7x\]
02

Combine like terms

Combine the terms on the right side of the equation. The like terms are \(-6x\) and \(7x\). This simplifies to: \[20 = (7x - 6x) - 3 \]or, \[20 = x - 3\]
03

Solve for x

Add \(3\) to both sides of the equation to isolate \(x\):\[20 + 3 = x - 3 + 3\] which simplifies to: \[23 = x\]
04

Verify the solution

Substitute \(x = 23\) back into the original equation to verify:Original equation:\[20 = -3(2(23) + 1) + 7(23)\]Simplify the terms inside the parentheses:\[20 = -3(46 + 1) + 161\]This reduces to:\[20 = -3(47) + 161 \]Calculate:\[20 = -141 + 161\] Simplify:\[20 = 20 \]The equation holds true, verifying our solution.

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Key Concepts

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

Distribution Property in Algebra
When solving linear equations, the distribution property plays a crucial role. This property enables us to multiply each term inside a set of parentheses by a number outside of them. It's often used to eliminate parentheses and simplify equations.
For example, consider the original equation:
  • \(20 = -3(2x + 1) + 7x\)
To apply the distribution property, multiply
  • \(-3\) by \(2x\)
  • \(-3\) by \(1\)
After distributing, you get:
  • \(-3(2x) = -6x\)
  • \(-3(1) = -3\)
So, the equation simplifies to:
  • \(20 = -6x - 3 + 7x\)
This step is essential because it transforms the equation into a format that is easier to work with.
Combining Like Terms
After applying the distribution property, the next step in solving the equation is to combine like terms. Like terms in an equation are terms that have the same variable raised to the same power. In our equation:
  • \(20 = -6x - 3 + 7x\)
we can see that \(-6x\) and \(7x\) are like terms because they both contain the variable \(x\) to the same power.
Combining these gives:
  • \((7x - 6x) = x\)
Thus, the equation becomes:
  • \(20 = x - 3\)
Combining like terms reduces the complexity of equations, making it easier to isolate the variable and solve for it in the subsequent steps.
Verification of Solutions
The final step in solving an equation is to verify the solution. This is important to ensure that the value you found for the variable satisfies the original equation. In our exercise, we found:
  • \(x = 23\)
To verify, substitute \(x = 23\) back into the original equation:
  • \(20 = -3(2(23) + 1) + 7(23)\)
Simplifying the equation gives:
  • \(20 = -3(47) + 161\)
Calculating further:
  • \(20 = -141 + 161\)
  • \(20 = 20\)
The final equality confirms that \(x = 23\) is indeed the correct solution. Verification not only gives you confidence in your answer but also reinforces your understanding of the manipulation involved in algebraic transformations.

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