Resuelve la siguiente ecuación.
$-\frac{1}{24} y+\frac{3}{2}-\frac{9}{14} y=-\frac{1}{12}+\frac{2}{3} y$
Answer (1)
Combine like terms by moving all terms containing y to one side and constants to the other side of the equation.
Find a common denominator for the fractions on each side and rewrite the fractions.
Add the fractions on both sides to simplify the equation to the form A y = B .
Solve for y by dividing both sides by A , resulting in y = − 227 266 .
Explanation
Analyzing the Equation First, let's analyze the given equation: − 24 1 y + 2 3 − 14 9 y = − 12 1 + 3 2 y Our goal is to isolate y on one side of the equation to find its value.
Grouping Like Terms Next, we'll group the terms containing y on one side and the constant terms on the other side. Let's move all y terms to the right side and constant terms to the left side: 2 3 + 12 1 = 3 2 y + 24 1 y + 14 9 y Now, we find a common denominator for the fractions on each side. On the left side, the common denominator for 2 and 12 is 12. On the right side, the common denominator for 3, 24, and 14 is 168.
Rewriting with Common Denominators Now, we rewrite the fractions with the common denominators: 2 × 6 3 × 6 + 12 1 = 3 × 56 2 × 56 y + 24 × 7 1 × 7 y + 14 × 12 9 × 12 y 12 18 + 12 1 = 168 112 y + 168 7 y + 168 108 y Now, we add the fractions on both sides: 12 19 = 168 227 y
Isolating y To solve for y , we multiply both sides by the reciprocal of 168 227 :
y = 12 19 × 227 168 Now, we simplify the fraction: y = 12 × 227 19 × 168 y = 1 × 227 19 × 14 y = 227 266 Since we moved the y terms to the right side, the sign is negative, so y = − 227 266
Final Answer Therefore, the solution to the equation is: y = − 227 266
Examples
When balancing chemical equations, you often need to solve linear equations similar to this one to find the stoichiometric coefficients. For example, if you have an equation like a A + b B = c C + d D , where a , b , c , d are the coefficients you need to find, and you have some constraints relating these coefficients, you might end up with an equation similar to the one we solved. Solving for one of the coefficients in terms of the others helps you balance the equation correctly, ensuring that the number of atoms of each element is the same on both sides of the equation. This is crucial in chemistry to accurately represent chemical reactions.
