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Chapter 2: Problem 9

Determine whether each equation is satisfied by the given number. $$3 x+7=-5, \quad-4$$

Short Answer

Expert verified
Yes, \(x = -4\) satisfies the equation \(3x + 7 = -5\).

Step by step solution

01

Substitute the Given Number into the Equation

Substitute \(x = -4\) into the equation \(3x + 7 = -5\). The equation becomes \(3(-4) + 7 = -5\).
02

Simplify the Left Side of the Equation

Multiply \(3 \times -4\) to get \(-12\). Then the equation reads \(-12 + 7 = -5\).
03

Combine Like Terms on the Left Side

Add \(-12 + 7\) to get \(-5\). So the equation now reads \(-5 = -5\) which is true.
04

Conclusion

Since both sides of the equation are equal, \(x = -4\) satisfies the equation \(3x + 7 = -5\).

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Key Concepts

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

substitution method
The substitution method is a simple and effective technique for solving linear equations. Essentially, you replace a variable in the equation with a given value to see if the equation holds true. For example, let's consider the equation \(3x + 7 = -5\). If we're given that \(x = -4\), we substitute \(-4\) in place of \(x\) in the equation. This turns our original equation into \(3(-4) + 7 = -5\). Substitution helps us to quickly determine whether a particular value satisfies the equation or not.
combining like terms
Combining like terms is a crucial step in solving linear equations. It involves simplifying an equation by consolidating terms that have the same variable. In our example, after substituting \(x = -4\) into the equation \(3x + 7 = -5\), the next step was to simplify the left side. This is done by multiplying \(3\) by \(-4\), resulting in \(-12\). Next, we add \(-12\) and \(7\). Since both are constants, they can be combined to give \(-5\). Simplifying equations in this way makes it easier to see whether an equation is satisfied.
satisfying equations
Finally, to determine if an equation is satisfied, both sides of the equation must be equal after substitution and simplification. Following our previous steps, we turned our equation into \(-5 = -5\). Because both sides of the equation are identical, we can conclude that the original value we substituted, \(x = -4\), satisfies the equation. This process confirms that the given value does indeed make the equation true, which is essential in verifying solutions.

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