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Magazine Chapter 6: Problem 43
Solve each equation. $$ \frac{-15}{4 y+1}+4=y $$
Short Answer
Expert verified
The solutions are \(y = \frac{11}{4}\) and \(y = 1\).
Step by step solution
01
Clear the Fraction
The first step to solve the equation \(\frac{-15}{4y+1} + 4 = y\) is to eliminate the fraction. Subtract 4 from both sides to isolate the fraction:\[\frac{-15}{4y+1} = y - 4\]
02
Cross Multiply
To eliminate the fraction, perform cross multiplication. Multiply both sides by \(4y + 1\):\[-15 = (y-4)(4y+1)\]
03
Expand the Expression
Expand the right side of the equation to remove the parentheses:\[-15 = y(4y) + y(1) - 4(4y) - 4(1)\]which simplifies to:\[-15 = 4y^2 + y - 16y - 4\]and then:\[-15 = 4y^2 - 15y - 4\]
04
Set the Equation to Zero
Rearrange the equation to set it to zero:\[4y^2 - 15y - 4 + 15 = 0\]This simplifies to:\[4y^2 - 15y + 11 = 0\]
05
Solve the Quadratic Equation
The equation \(4y^2 - 15y + 11 = 0\) is quadratic. Use the quadratic formula \(y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) where \(a = 4\), \(b = -15\), and \(c = 11\). Calculate the discriminant first:\[b^2 - 4ac = (-15)^2 - 4(4)(11) = 225 - 176 = 49\]Now solve for \(y\):\[y = \frac{-(-15) \pm \sqrt{49}}{2(4)} = \frac{15 \pm 7}{8}\]This gives us two solutions: \(y = \frac{22}{8} = \frac{11}{4}\) and \(y = \frac{8}{8} = 1\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Quadratic Formula
Understanding the quadratic formula is a key skill in solving quadratic equations. A quadratic equation is one that can be written in the form of \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants, and \(x\) represents the variable. The quadratic formula is given by:\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]This formula allows us to find the value of \(x\) by performing arithmetic on the coefficients of the equation. To use the quadratic formula:
- Identify \(a\), \(b\), and \(c\) in your quadratic equation.
- Substitute these values into the quadratic formula.
- Calculate the discriminant \(b^2 - 4ac\) to determine the nature of the roots.
- Solve the formula, which may result in two possible values due to the \(\pm\) symbol.
Cross Multiplication
Cross multiplication is a useful technique for clearing fractions from rational equations. When you're faced with an equation involving a fraction, you can "cross multiply" to simplify and find a solution. Here's how it works:
- Consider an equation of the form \(\frac{a}{b} = c\), cross multiplying involves multiplying \(a\) by the denominator of the term on the opposite side of the equation, in this case, multiplying by \(b \cdot 1\).
- Perform the same operation on the other side of the equation. This gives \(a = bc\).
- This technique essentially eliminates the fraction, turning a rational equation into a linear or quadratic equation.
Discriminant
The discriminant is a critical part of the quadratic formula and aids in understanding the nature of the roots of a quadratic equation. It is represented as \(b^2 - 4ac\) in the quadratic formula.The discriminant helps in predicting:
- If the discriminant > 0: The quadratic equation has two distinct real roots.
- If the discriminant = 0: There is exactly one real root, also known as a repeated or double root.
- If the discriminant < 0: The equation has no real roots and the solutions are complex or imaginary.
