Solve the equation and check the solution. Express numbers as integers or simplified fractions.
[tex]2-4(y-4)=5(2 y+8)[/tex]
The solution set is $\square$.
Answer (2)
The equation 2 − 4 ( y − 4 ) = 5 ( 2 y + 8 ) is solved to find that y = − 7 11 . The solution is verified by substituting y back into the original equation, confirming the left side equals the right side. Therefore, the solution set is {\frac{-11}{7}}.
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Expand both sides of the equation: 2 − 4 ( y − 4 ) = 5 ( 2 y + 8 ) becomes 18 − 4 y = 10 y + 40 .
Isolate the variable y by adding 4 y to both sides and subtracting 40 from both sides: − 22 = 14 y .
Solve for y by dividing both sides by 14: y = 14 − 22 .
Simplify the fraction: y = − 7 11 .
Explanation
Expanding the equation We are given the equation 2 − 4 ( y − 4 ) = 5 ( 2 y + 8 ) and we need to solve for y . First, we will expand both sides of the equation.
Expanding both sides Expanding the left side, we have: 2 − 4 ( y − 4 ) = 2 − 4 y + 16 = 18 − 4 y Expanding the right side, we have: 5 ( 2 y + 8 ) = 10 y + 40 So the equation becomes: 18 − 4 y = 10 y + 40
Isolating the variable Now, we want to isolate the variable y . Let's add 4 y to both sides of the equation: 18 − 4 y + 4 y = 10 y + 40 + 4 y 18 = 14 y + 40 Next, subtract 40 from both sides: 18 − 40 = 14 y + 40 − 40 − 22 = 14 y
Solving for y Now, we solve for y by dividing both sides by 14: 14 − 22 = 14 14 y y = 14 − 22 We can simplify the fraction by dividing both the numerator and the denominator by 2: y = 7 − 11
Checking the solution Now, we check our solution by substituting y = 7 − 11 back into the original equation: 2 − 4 ( 7 − 11 − 4 ) = 5 ( 2 ( 7 − 11 ) + 8 ) 2 − 4 ( 7 − 11 − 7 28 ) = 5 ( 7 − 22 + 7 56 ) 2 − 4 ( 7 − 39 ) = 5 ( 7 34 ) 2 + 7 156 = 7 170 7 14 + 7 156 = 7 170 7 170 = 7 170 The solution checks out.
Final Answer Therefore, the solution set is { 7 − 11 } .
Examples
In real-world scenarios, solving linear equations like this can be applied to various problems. For example, imagine you are trying to determine the break-even point for a small business. You have fixed costs and variable costs, and you want to find out how many units you need to sell to cover all your expenses. By setting up a linear equation that represents your costs and revenue, you can solve for the number of units needed to break even. This helps in making informed business decisions and financial planning. The equation we solved is a simplified version of such a scenario, where understanding the value of 'y' helps in balancing different factors.
