Solve For $y$ If:$(y+3)(4y+9) = 2y+15$INSTRUCTION: Give Your Answers As Simplified Fractions.Answer: $ Y = Y= Y = [/tex] $\square$ Or $y=$ $ □ \square □ [/tex]
Introduction
In this article, we will focus on solving for y in the given equation (y+3)(4y+9) = 2y+15. This equation involves the multiplication of two binomials, and we will use the distributive property to simplify and solve for y.
Step 1: Expand the Left Side of the Equation
To solve for y, we first need to expand the left side of the equation using the distributive property. The distributive property states that for any real numbers a, b, and c, a(b+c) = ab + ac.
(y+3)(4y+9) = y(4y+9) + 3(4y+9)
Step 2: Simplify the Left Side of the Equation
Now, we can simplify the left side of the equation by multiplying the terms.
y(4y+9) + 3(4y+9) = 4y^2 + 9y + 12y + 27
Step 3: Combine Like Terms
Next, we can combine like terms on the left side of the equation.
4y^2 + 9y + 12y + 27 = 4y^2 + 21y + 27
Step 4: Set the Equation Equal to Zero
Now, we can set the equation equal to zero by subtracting 2y+15 from both sides.
4y^2 + 21y + 27 - (2y+15) = 0
Step 5: Simplify the Right Side of the Equation
Next, we can simplify the right side of the equation by combining like terms.
4y^2 + 21y + 27 - 2y - 15 = 4y^2 + 19y + 12
Step 6: Factor the Quadratic Equation
Now, we can factor the quadratic equation on the left side of the equation.
4y^2 + 19y + 12 = (4y+3)(y+4)
Step 7: Solve for y
Finally, we can solve for y by setting each factor equal to zero.
4y+3 = 0 \quad \text{or} \quad y+4 = 0
Solving the First Equation
To solve the first equation, we can subtract 3 from both sides and then divide by 4.
4y+3 = 0 \quad \Rightarrow \quad 4y = -3 \quad \Rightarrow \quad y = -\frac{3}{4}
Solving the Second Equation
To solve the second equation, we can subtract 4 from both sides.
y+4 = 0 \quad \Rightarrow \quad y = -4
Conclusion
In this article, we solved for y in the given equation (y+3)(4y+9) = 2y+15. We used the distributive property to expand the left side of the equation, combined like terms, and factored the quadratic equation. Finally, we solved for y by setting each factor equal to zero. The solutions to the equation are y = -\frac{3}{4} and y = -4.
Final Answer
Introduction
In our previous article, we solved for y in the given equation (y+3)(4y+9) = 2y+15. We used the distributive property to expand the left side of the equation, combined like terms, and factored the quadratic equation. In this article, we will answer some frequently asked questions about solving for y in the given equation.
Q: What is the first step in solving for y in the given equation?
A: The first step in solving for y in the given equation is to expand the left side of the equation using the distributive property.
Q: How do I expand the left side of the equation?
A: To expand the left side of the equation, you need to multiply the terms inside the parentheses. For example, (y+3)(4y+9) = y(4y+9) + 3(4y+9).
Q: What is the next step after expanding the left side of the equation?
A: After expanding the left side of the equation, you need to combine like terms. This involves adding or subtracting the coefficients of the same variables.
Q: How do I combine like terms?
A: To combine like terms, you need to add or subtract the coefficients of the same variables. For example, 4y^2 + 9y + 12y + 27 = 4y^2 + 21y + 27.
Q: What is the next step after combining like terms?
A: After combining like terms, you need to set the equation equal to zero by subtracting 2y+15 from both sides.
Q: How do I set the equation equal to zero?
A: To set the equation equal to zero, you need to subtract 2y+15 from both sides of the equation. For example, 4y^2 + 21y + 27 - (2y+15) = 0.
Q: What is the next step after setting the equation equal to zero?
A: After setting the equation equal to zero, you need to simplify the right side of the equation by combining like terms.
Q: How do I simplify the right side of the equation?
A: To simplify the right side of the equation, you need to combine like terms. For example, 4y^2 + 21y + 27 - 2y - 15 = 4y^2 + 19y + 12.
Q: What is the next step after simplifying the right side of the equation?
A: After simplifying the right side of the equation, you need to factor the quadratic equation on the left side of the equation.
Q: How do I factor the quadratic equation?
A: To factor the quadratic equation, you need to find two numbers whose product is the constant term and whose sum is the coefficient of the linear term. For example, 4y^2 + 19y + 12 = (4y+3)(y+4).
Q: What is the final step in solving for y in the given equation?
A: The final step in solving for y in the given equation is to solve for y by setting each factor equal to zero.
Q: How do I solve for y?
A: To solve for y, you need to set each factor equal to zero and solve for y. For example, 4y+3 = 0 or y+4 = 0.
Conclusion
In this article, we answered some frequently asked questions about solving for y in the given equation (y+3)(4y+9) = 2y+15. We covered the steps involved in solving for y, including expanding the left side of the equation, combining like terms, setting the equation equal to zero, simplifying the right side of the equation, factoring the quadratic equation, and solving for y.
Final Answer
The final answer is or .