Solve for the variable in the following equation:
[tex]$\frac{8}{9}-8 y-\frac{2}{3}=\frac{4}{3}+\frac{2}{3} y+\frac{3}{8} y$[/tex]
Answer (1)
Combine the constant terms on both sides of the equation: 9 2 − 8 y = 3 4 + 3 2 y + 8 3 y .
Combine the y terms on both sides of the equation: 9 2 − 8 y = 3 4 + 24 25 y .
Isolate y on one side of the equation: − 9 10 = 24 217 y .
Solve for y : y = − 651 80 .
Explanation
Problem Analysis We are given the equation 9 8 − 8 y − 3 2 = 3 4 + 3 2 y + 8 3 y Our goal is to solve for y .
Simplifying Constants First, let's simplify the constant terms on both sides of the equation. On the left side, we have 9 8 − 3 2 . Since 3 2 = 9 6 , we have 9 8 − 9 6 = 9 2 . So the left side becomes 9 2 − 8 y . On the right side, we have just 3 4 . Thus, the equation becomes 9 2 − 8 y = 3 4 + 3 2 y + 8 3 y
Combining y Terms Next, let's combine the y terms on the right side of the equation. We have 3 2 y + 8 3 y . To add these, we need a common denominator, which is 24. So we have 3 2 = 24 16 and 8 3 = 24 9 . Thus, 24 16 y + 24 9 y = 24 25 y . The equation now becomes 9 2 − 8 y = 3 4 + 24 25 y
Isolating y Terms Now, let's move all the y terms to one side and the constant terms to the other side. Add 8 y to both sides and subtract 3 4 from both sides: 9 2 − 3 4 = 24 25 y + 8 y To combine the constants on the left, we need a common denominator, which is 9. Since 3 4 = 9 12 , we have 9 2 − 9 12 = − 9 10 . To combine the y terms on the right, we need a common denominator, which is 24. Since 8 = 24 8 × 24 = 24 192 , we have 24 25 y + 24 192 y = 24 217 y . Thus, the equation becomes − 9 10 = 24 217 y
Solving for y Finally, to solve for y , we multiply both sides by 217 24 :
y = − 9 10 × 217 24 = − 9 × 217 10 × 24 = − 1953 240 We can simplify this fraction by dividing both numerator and denominator by 3: y = − 651 80 Thus, the solution is y = − 651 80 .
Final Answer Therefore, the solution to the equation is y = − 651 80
Examples
When you are trying to determine the amount of ingredients to use in a recipe, you might need to solve a linear equation like this one. For example, if you know the total amount of a mixture and the proportion of each ingredient, you can set up an equation to find the exact quantity of each ingredient needed. This is also applicable in chemistry when balancing chemical equations or in physics when calculating forces and motion.
