Solve the equation.
[tex]\frac{x+6}{2}=5-\frac{x+4}{5}[/tex]
What is the solution? Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
A. The solution set is $\square$ (Type an integer or a fraction.)
B. The solution set is \{x \mid x$ is a real number \}.
C. The solution set is $\varnothing$.
Answer (2)
Multiply both sides by 10 to eliminate fractions: 5 ( x + 6 ) = 50 − 2 ( x + 4 ) .
Expand and simplify: 5 x + 30 = 42 − 2 x .
Isolate x: 7 x = 12 .
Solve for x: x = 7 12 .
Explanation
Problem Setup We are given the equation 2 x + 6 = 5 − 5 x + 4 and we want to solve for x .
Eliminating Fractions To eliminate the fractions, we multiply both sides of the equation by the least common multiple of the denominators, which is 10: 10 ( 2 x + 6 ) = 10 ( 5 − 5 x + 4 ) Simplifying, we get 5 ( x + 6 ) = 50 − 2 ( x + 4 ) .
Expanding and Simplifying Expanding both sides of the equation, we have 5 x + 30 = 50 − 2 x − 8 Combining like terms on the right side, we get 5 x + 30 = 42 − 2 x .
Isolating the Variable Now, we add 2 x to both sides of the equation: 5 x + 2 x + 30 = 42 − 2 x + 2 x 7 x + 30 = 42 Subtracting 30 from both sides, we have 7 x + 30 − 30 = 42 − 30 7 x = 12 .
Solving for x Finally, we divide both sides by 7 to solve for x :
7 7 x = 7 12 x = 7 12 Thus, the solution set is { 7 12 } .
Final Answer The solution set is { 7 12 } . Therefore, the correct choice is A, and we fill in the box with 7 12 .
Examples
Imagine you're baking a cake and need to adjust the ingredient quantities based on the number of servings. Solving linear equations like this helps you determine the exact amount of each ingredient needed to maintain the recipe's balance. For example, if the original recipe serves 6 and you need to serve 10, you can set up a linear equation to find the adjusted quantities. This ensures your cake turns out perfectly, no matter the serving size. Linear equations are essential for scaling recipes and maintaining consistent results in cooking.
The solution to the equation 2 x + 6 = 5 − 5 x + 4 is x = 7 12 . Therefore, the correct choice is A, and we fill in the box with 7 12 .
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