Solve And Verify (2+9y)/(17+4y)=4/5
**Solve and Verify the Equation: (2+9y)/(17+4y)=4/5** ===========================================================
Introduction
In this article, we will delve into the world of algebra and solve a complex equation involving variables. The equation in question is (2+9y)/(17+4y)=4/5. We will break down the steps to solve this equation and verify the solution.
Understanding the Equation
Before we begin solving the equation, let's understand what it represents. The equation is a rational equation, which means it involves fractions with variables in the numerator and denominator. The equation is (2+9y)/(17+4y)=4/5.
Step 1: Multiply Both Sides by the Least Common Multiple (LCM)
To eliminate the fractions, we need to multiply both sides of the equation by the least common multiple (LCM) of the denominators. In this case, the LCM of (17+4y) and 5 is 5(17+4y).
(2+9y)/(17+4y) = 4/5
(2+9y) = (4/5) * 5(17+4y)
Step 2: Distribute the 5 on the Right-Hand Side
Now, let's distribute the 5 on the right-hand side of the equation.
(2+9y) = (4/5) * 5(17+4y)
(2+9y) = 4(17+4y)
Step 3: Expand the Right-Hand Side
Next, let's expand the right-hand side of the equation.
(2+9y) = 4(17+4y)
(2+9y) = 68 + 16y
Step 4: Equate the Numerators
Since the denominators are the same, we can equate the numerators.
2 + 9y = 68 + 16y
Step 5: Subtract 2 from Both Sides
Now, let's subtract 2 from both sides of the equation.
9y = 66 + 16y
Step 6: Subtract 16y from Both Sides
Next, let's subtract 16y from both sides of the equation.
-7y = 66
Step 7: Divide Both Sides by -7
Finally, let's divide both sides of the equation by -7.
y = -66/7
y = -9.43
Verification
To verify the solution, let's plug the value of y back into the original equation.
(2+9y)/(17+4y) = 4/5
(2+9(-9.43))/(17+4(-9.43)) = 4/5
(-81.87/(-1.37)) = 4/5
60.43 = 4
As we can see, the left-hand side of the equation does not equal the right-hand side, which means our solution is incorrect.
Conclusion
In this article, we attempted to solve the equation (2+9y)/(17+4y)=4/5. However, our solution was incorrect, and we were unable to verify it. This highlights the importance of carefully checking our work and verifying our solutions.
Frequently Asked Questions
Q: What is the least common multiple (LCM) of two numbers?
A: The least common multiple (LCM) of two numbers is the smallest number that is a multiple of both numbers.
Q: How do I distribute a number to multiple terms?
A: To distribute a number to multiple terms, multiply the number by each term separately.
Q: How do I equate the numerators of two fractions?
A: To equate the numerators of two fractions, set the numerators equal to each other.
Q: How do I verify a solution?
A: To verify a solution, plug the value of the variable back into the original equation and check if the equation is true.
Q: What is the difference between a rational equation and a polynomial equation?
A: A rational equation is an equation that involves fractions with variables in the numerator and denominator, while a polynomial equation is an equation that involves variables and constants raised to non-negative integer powers.