Problem 64
Question
Solve: \(7 x=10+6(11-2 x)\) (Section \(2.3,\) Example 3 )
Step-by-Step Solution
Verified Answer
The solution for 'x' is approximately \(x = \frac{64}{7}\).
1Step 1: Simplify the equation
Simplify the right side of the equation by first multiplying within the parenthesis, then add or subtract as needed. This leads to the equation: \(7 x = 10 + 6 \times 9\)
2Step 2: Further simplification
Simplify further, by performing the multiplication on the right side, resulting in: \(7 x = 10 + 54\)
3Step 3: Perform addition
The equation after performing the addition on the right side becomes: \(7 x = 64\)
4Step 4: Solve for 'x'
Finally, the value of 'x' can be obtained by dividing both sides by 7. This leads to the solution: \(x = \frac{64}{7}\)
Key Concepts
Algebraic ManipulationSimplifying ExpressionsSolving for a Variable
Algebraic Manipulation
When tackling a linear equation, you'll often employ algebraic manipulation techniques. These methods involve rearranging, adding, subtracting, multiplying, or dividing parts of an equation to isolate the variable you're solving for. Let's explore how this works.
In the exercise, you encounter the equation: \( 7x = 10 + 6(11 - 2x) \). To manipulate this algebraically means that you need to make the equation simpler or express it in a form that isolates the variable \( x \).
Start by addressing terms inside the parentheses. Apply the distributive property, \( a(b + c) = ab + ac \), which helps in opening brackets by multiplying each term inside the parenthesis by the number outside.
Sequentially applying such steps simplifies the equation, making the variable eventually more accessible to solve for. Remember, balancing both sides is key, so any operation applied on one side should be equally performed on the other.
In the exercise, you encounter the equation: \( 7x = 10 + 6(11 - 2x) \). To manipulate this algebraically means that you need to make the equation simpler or express it in a form that isolates the variable \( x \).
Start by addressing terms inside the parentheses. Apply the distributive property, \( a(b + c) = ab + ac \), which helps in opening brackets by multiplying each term inside the parenthesis by the number outside.
Sequentially applying such steps simplifies the equation, making the variable eventually more accessible to solve for. Remember, balancing both sides is key, so any operation applied on one side should be equally performed on the other.
Simplifying Expressions
Simplifying expressions involves reducing an equation to its simplest form. This process makes the expression more manageable and the solving process straightforward.
In this problem, after distributing, you need to simplify \( 6(11 - 2x) \) by calculating each term individually to simplify the right side. Multiply 6 by both 11 and \(-2x\) yielding \( 66 - 12x \).
Next, perform the arithmetic operations, such as addition and subtraction, to further simplify the expression. Combining like terms in equations minimizes the complexity, focusing on significant numbers only. For instance, adding or subtracting constants maintains balance on both sides of an equation while simplifying them.
In this problem, after distributing, you need to simplify \( 6(11 - 2x) \) by calculating each term individually to simplify the right side. Multiply 6 by both 11 and \(-2x\) yielding \( 66 - 12x \).
Next, perform the arithmetic operations, such as addition and subtraction, to further simplify the expression. Combining like terms in equations minimizes the complexity, focusing on significant numbers only. For instance, adding or subtracting constants maintains balance on both sides of an equation while simplifying them.
- Ensure any numbers added or subtracted are combined to clean up the expression.
- Keep numbers attached to variables unevolved until necessary.
Solving for a Variable
The ultimate goal in solving linear equations is to find the value of the variable, often referred to as "solving for \( x \)." Achieving this involves breaking down the equation until the variable is isolated on one side of the equation.
After simplifying \( 7x = 10 + 54 \), you solve for \( x \) by isolating it. This often requires undoing operations: in this case, the multiplication of \( 7 \) and \( x \).
To do this, divide both sides of the equation by 7, following one crucial rule: whatever you do to one side, do the same to the other to maintain equation balance.
As a result, dividing gives \( x = \frac{64}{7} \). Remember: always check your answer by substituting it back into the original equation to verify that it makes both sides equal. Finding the variable is the reward of careful simplification and manipulation steps.
After simplifying \( 7x = 10 + 54 \), you solve for \( x \) by isolating it. This often requires undoing operations: in this case, the multiplication of \( 7 \) and \( x \).
To do this, divide both sides of the equation by 7, following one crucial rule: whatever you do to one side, do the same to the other to maintain equation balance.
As a result, dividing gives \( x = \frac{64}{7} \). Remember: always check your answer by substituting it back into the original equation to verify that it makes both sides equal. Finding the variable is the reward of careful simplification and manipulation steps.
Other exercises in this chapter
Problem 64
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