Problem 73
Question
Solve each equation. \(2 x-7=3(x-4)\)
Step-by-Step Solution
Verified Answer
The solution to the equation is \(x = 5\).
1Step 1: Distribute on the Right Side
Distribute the 3 to both terms inside the parenthesis on the right side of the equation:The equation becomes:\[ 2x - 7 = 3x - 12 \]
2Step 2: Move Terms with x to One Side
Subtract \(3x\) from both sides of the equation to move all terms containing \(x\) to one side:The equation becomes:\[ 2x - 3x - 7 = -12 \]
3Step 3: Simplify the Equation
Combine the \(x\) terms on the left side:The equation simplifies to:\[ -x - 7 = -12 \]
4Step 4: Isolate the x Term
Add 7 to both sides to isolate the term containing \(x\):This changes the equation to:\[ -x = -5 \]
5Step 5: Solve for x
Multiply both sides of the equation by -1 to solve for \(x\):You get:\[ x = 5 \]
Key Concepts
Equation DistributionCombining Like TermsIsolating VariablesInverse Operations
Equation Distribution
One of the first steps in solving linear equations is applying the distribution of numbers across parentheses. In our example, we encounter the equation \( 2x - 7 = 3(x - 4) \). Here, the number 3 needs to be distributed across each term inside the parentheses.
Think of distribution as multiplying the number outside the parentheses by each term inside. In our scenario, the 3 should be applied to both 'x' and '-4', which results in \( 3 \times x + 3 \times (-4) = 3x -12 \).
This step transforms the equation into \( 2x - 7 = 3x - 12 \). Distribution is crucial because it simplifies equations by removing the parentheses and preparing it for further steps like combining terms.
Think of distribution as multiplying the number outside the parentheses by each term inside. In our scenario, the 3 should be applied to both 'x' and '-4', which results in \( 3 \times x + 3 \times (-4) = 3x -12 \).
This step transforms the equation into \( 2x - 7 = 3x - 12 \). Distribution is crucial because it simplifies equations by removing the parentheses and preparing it for further steps like combining terms.
Combining Like Terms
When solving equations, combining like terms helps simplify the expression. In our example, after distributing, you're left with:\( 2x - 7 = 3x - 12 \). Here, we need to reorganize it so that all the 'x' terms are on one side.
To achieve this, subtract \( 3x \) from both sides to bring all similar terms together:\( 2x - 3x - 7 = -12 \).
When you combine \( 2x \) and \(-3x \), you will get \(-x \). This simplified expression \(-x - 7 = -12 \) is much easier to work with. Combining like terms is a basic yet essential skill in algebra for making equations simpler and easier to solve.
To achieve this, subtract \( 3x \) from both sides to bring all similar terms together:\( 2x - 3x - 7 = -12 \).
When you combine \( 2x \) and \(-3x \), you will get \(-x \). This simplified expression \(-x - 7 = -12 \) is much easier to work with. Combining like terms is a basic yet essential skill in algebra for making equations simpler and easier to solve.
Isolating Variables
After combining like terms, the next mission is isolating the variable (in this case, \(x\)). This means getting \(x\) on one side of the equation, and all the constants on the other.
We're looking at the equation: \(-x - 7 = -12 \). To isolate \(-x\), add 7 to both sides, which cancels out the \(-7\) on the left side:
\(-x = -5 \).
Isolating the variable is essential as it sets up the equation for the last steps towards solving for the variable's value.
We're looking at the equation: \(-x - 7 = -12 \). To isolate \(-x\), add 7 to both sides, which cancels out the \(-7\) on the left side:
\(-x = -5 \).
Isolating the variable is essential as it sets up the equation for the last steps towards solving for the variable's value.
Inverse Operations
The final piece of the puzzle in solving equations is applying inverse operations, which is simply undoing what's done to the variable.
Since the isolated term is \(-x\), the inverse operation to clear the negative sign is multiplying both sides by \(-1\).
This gives us:\(x = 5\). Using inverse operations ensures you've reversed any operations previously done on the equation, ultimately providing the solution.
Recognizing when to use inverse operations can make solving equations efficient and straightforward, as it directly leads us to the correct answer.
Since the isolated term is \(-x\), the inverse operation to clear the negative sign is multiplying both sides by \(-1\).
This gives us:\(x = 5\). Using inverse operations ensures you've reversed any operations previously done on the equation, ultimately providing the solution.
Recognizing when to use inverse operations can make solving equations efficient and straightforward, as it directly leads us to the correct answer.
Other exercises in this chapter
Problem 72
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