Q. \[ \dfrac{9}{4}y – 12 = \dfrac{1}{4}y – 4 \]

Q: What is the solution to \dfrac{9}{4}y-12=\dfrac{1}{4}y-4?

Multiply by 4: 9y-48=y-16. Then 8y=32, so y=4.

Q: How do I check that y = 4 is correct?

Substitute: \dfrac{9}{4}(4)-12=9-12=-3 and \dfrac{1}{4}(4)-4=1-4=-3. Both sides equal, so y=4 is correct.

Q: Why multiply both sides by 4?

Multiplying by 4 clears denominators, turning \dfrac{9}{4}y and \dfrac{1}{4}y into integers, which simplifies arithmetic and reduces error.

Q: Can I solve without clearing denominators first?.

Yes: subtract \dfrac{1}{4}y from both sides to get \left(\dfrac{8}{4}\right)y-12=-4 , i.e. 2y-12=-4 , then 2y=8 , y=4 .

Q: How would I convert the equation to decimals?

Replace fractions: \dfrac{9}{4}=2.25 and \dfrac{1}{4}=0.25, giving 2.25y-12=0.25y-4, then solve: 2y=8, y=4..

Q: How does this relate to graphing?

Each side is a linear function of y treated as a number; solving finds the single y where the two expressions are equal. If plotted vs y, their intersection occurs at y=4.

Answer

Start with the equation.

\(\frac{9}{4}y-12=\frac{1}{4}y-4\)

Subtract \(\frac{1}{4}y\) from both sides.

\(\frac{9}{4}y-\frac{1}{4}y-12=-4\)

Simplify the \(y\)-terms.

\(\frac{8}{4}y-12=-4\)

\(2y-12=-4\)

Add \(12\) to both sides.

\(2y=8\)

Divide both sides by \(2\).

\(y=4\)

Final result: \(y=4\)

Detailed Explanation

Step-by-step solution

Write the given equation.
\[ \tfrac{9}{4}y – 12 = \tfrac{1}{4}y – 4 \]
To eliminate the fractions, multiply both sides of the equation by 4. Multiplying both sides by the same nonzero number preserves the equality.
\[ 4\bigg(\tfrac{9}{4}y – 12\bigg) = 4\bigg(\tfrac{1}{4}y – 4\bigg) \]
Distribute 4 on each side and simplify the fractions:
\[ 9y – 48 = y – 16 \]

Explanation: 4 times 9/4 is 9, 4 times 12 is 48, 4 times 1/4 is 1, and 4 times 4 is 16.
Collect the variable terms on one side by subtracting y from both sides (subtracting the same quantity from both sides preserves equality):
\[ 9y – 48 – y = y – 16 – y \]

Simplify:

\[ 8y – 48 = -16 \]
Isolate the term with y by adding 48 to both sides (adding the same number to both sides preserves equality):
\[ 8y – 48 + 48 = -16 + 48 \]

Simplify:

\[ 8y = 32 \]
Divide both sides by 8 to solve for y (dividing both sides by the same nonzero number preserves equality):
\[ y = \tfrac{32}{8} \]

Simplify:

\[ y = 4 \]
Verify the solution by substituting y = 4 back into the original equation:
Left side: \[ \tfrac{9}{4}\cdot 4 – 12 = 9 – 12 = -3 \]

Right side: \[ \tfrac{1}{4}\cdot 4 – 4 = 1 – 4 = -3 \]

Both sides are equal, so y = 4 is correct.

Final answer: \[ y = 4 \]

See full solution

Struggling with math? Try our AI homework help tools for free!

AI homework helper

Algebra FAQs

\(What is the solution to \(\dfrac{9}{4}y-12=\dfrac{1}{4}y-4\)?\)

Multiply by 4: \(9y-48=y-16\). Then \(8y=32\), so \(y=4\).

How do I check that \( y = 4 \) is correct?

Substitute: \(\dfrac{9}{4}(4)-12=9-12=-3\) and \(\dfrac{1}{4}(4)-4=1-4=-3\). Both sides equal, so \(y=4\) is correct.

Why multiply both sides by \(4\)?

Multiplying by 4 clears denominators, turning \( \dfrac{9}{4}y \) and \( \dfrac{1}{4}y \) into integers, which simplifies arithmetic and reduces error.

Can I solve without clearing denominators first?.

Yes: subtract \( \dfrac{1}{4}y \) from both sides to get \( \left(\dfrac{8}{4}\right)y-12=-4 \), i.e. \( 2y-12=-4 \), then \( 2y=8 \), \( y=4 \).

What common mistakes should I avoid?

Forgetting to multiply every term when clearing fractions, sign errors when moving terms, and arithmetic mistakes when combining coefficients are typical pitfalls.

What if the variable coefficients were equal on both sides?

How would I convert the equation to decimals?

Replace fractions: \(\dfrac{9}{4}=2.25\) and \(\dfrac{1}{4}=0.25\), giving \(2.25y-12=0.25y-4\), then solve: \(2y=8\), \(y=4\)..

How does this relate to graphing?

Each side is a linear function of \(y\) treated as a number; solving finds the single \(y\) where the two expressions are equal. If plotted vs \(y\), their intersection occurs at \(y=4\).

Solve math problems with our AI tools
Try our economics and finance help

252,312+ customers tried
Analytical, General, Biochemistry, etc.

Finance homework help Economics homework help Accounting homework help