GSEB Class 8 Maths Solutions Chapter 2 એકચલ સુરેખ સમીકરણ Exercise 2.6

Solve The Following Equations: (Examples 1 To 5)

Question 1. \( \frac{8 x-3}{3 x} = 2 \)
Answer:
\( \frac{8 x-3}{3 x} = 2 \)
We multiply both sides by \( 3x \).
\( 3x \left( \frac{8 x-3}{3 x} \right) = 3x(2) \)
\( 8x - 3 = 6x \)
Now, we move \( -3 \) to the right side and \( 6x \) to the left side.
\( 8x - 6x = 3 \)
\( 2x = 3 \)
We divide both sides by 2.
\( \frac{2 x}{2} = \frac{3}{2} \)
\( x = \frac {3}{2} \)
In simple words: To find the value of x, we first multiply both sides by 3x. Then, we gather the x terms on one side and constant numbers on the other side. Finally, we divide to get the answer.

Exam Tip: Remember to perform operations like multiplication or division on both sides of the equation to maintain balance and solve for the variable effectively.

Question 2. \( \frac{9 x}{7-6 x} = 2 \)
Answer:
\( \frac{9 x}{7-6 x} = 2 \)
We perform cross-multiplication here.
\( 9x = 2(7 - 6x) \)
\( 9x = 14 - 12x \)
Next, we move \( -12x \) to the left side.
\( 9x + 12x = 14 \)
\( 21x = 14 \)
We divide both sides by 21.
\( \frac{21 x}{21} = \frac{14}{21} \)
\( x = \frac {2}{3} \)
In simple words: Start by multiplying across the equals sign. Then, get all the x terms together on one side and the numbers on the other. Finish by dividing to find x.

Exam Tip: When cross-multiplying, make sure to distribute the number outside the bracket to all terms inside it to prevent common errors.

Question 3. \( \frac{z}{z+15}=\frac{4}{9} \)
Answer:
\( \frac{z}{z+15}=\frac{4}{9} \)
Let's cross-multiply the terms.
\( 9z = 4(z + 15) \)
\( 9z = 4z + 60 \)
Now, we move \( 4z \) to the left side.
\( 9z - 4z = 60 \)
\( 5z = 60 \)
We divide both sides by 5.
\( \frac{5 z}{5} = \frac{60}{5} \)
\( z = 12 \)
In simple words: To solve for z, first multiply diagonally across the equal sign. Then, bring all the z parts to one side and numbers to the other. Finally, divide to get the value of z.

Exam Tip: Always double-check your distribution when multiplying a number into a bracket to avoid calculation errors.

Question 4. \( \frac{3 y+4}{2-6 y}=\frac{-2}{5} \)
Answer:
\( \frac{3 y+4}{2-6 y}=\frac{-2}{5} \)
We perform cross-multiplication.
\( 5(3y + 4) = -2(2 - 6y) \)
\( 15y + 20 = -4 + 12y \)
Now, we move \( 20 \) to the right side and \( 12y \) to the left side.
\( 15y - 12y = -4 - 20 \)
\( 3y = -24 \)
We divide both sides by 3.
\( \frac{3 y}{3} = \frac{-24}{3} \)
\( y = -8 \)
In simple words: Start by cross-multiplying. Then, simplify both sides and gather the 'y' terms on one side and constant numbers on the other. Finally, divide to find 'y'.

Exam Tip: Pay close attention to negative signs during distribution and when moving terms across the equals sign; a common mistake is losing track of them.

Question 5. \( \frac{7 y+4}{y+2}=\frac{-4}{3} \)
Answer:
\( \frac{7 y+4}{y+2}=\frac{-4}{3} \)
Let's cross-multiply.
\( 3(7y + 4) = -4(y + 2) \)
\( 21y + 12 = -4y - 8 \)
Next, we move \( -4y \) to the left side and \( 12 \) to the right side.
\( 21y + 4y = -8 - 12 \)
\( 25y = -20 \)
We divide both sides by 25.
\( y = \frac{-20}{25} \)
\( y = \frac {-4}{5} \)
In simple words: Begin by multiplying diagonally. Expand both sides. Bring the 'y' terms to one side and the numbers to the other. Then, simplify and divide to get the answer.

Exam Tip: Always reduce fractions to their simplest form as the final step in the solution to ensure full marks.

Question 6. The current age ratio of Hari and Harry is 5 : 7. After 4 years, their age ratio will be 3 : 4. Find their current ages.
Answer:
The current age ratio of Hari and Harry is 5 : 7.
Let Hari's current age be \( 5x \) years.
And Harry's current age be \( 7x \) years.
After 4 years, Hari's age will be \( (5x + 4) \) years.
After 4 years, Harry's age will be \( (7x + 4) \) years.
After 4 years, the age ratio of Hari and Harry will be 3 : 4.
So, \( (5x + 4) : (7x + 4) = 3 : 4 \)
This can be written as:
\( \frac{5 x+4}{7 x+4} = \frac{3}{4} \)
We perform cross-multiplication.
\( 4(5x + 4) = 3(7x + 4) \)
\( 20x + 16 = 21x + 12 \)
Now, we move \( 21x \) to the left side and \( 16 \) to the right side.
\( 20x - 21x = 12 - 16 \)
\( -x = -4 \)
We multiply both sides by \( -1 \).
\( x = 4 \)
Therefore, Hari's current age is \( 5x = 5 \times 4 = 20 \) years.
And Harry's current age is \( 7x = 7 \times 4 = 28 \) years.
Thus, Hari's current age is 20 years and Harry's current age is 28 years.
In simple words: We used the given age ratio to set up expressions for their current ages with 'x'. Then, we formed a new ratio equation for their ages after 4 years. Solving this equation gave us the value of 'x', which helped us find their actual current ages.

Exam Tip: When dealing with age problems, always define variables for current ages first, then create expressions for future or past ages before setting up the equation.

Question 7. The denominator of a fraction is 8 more than its numerator. If 17 is added to the numerator and 1 is subtracted from the denominator, the new fraction becomes \( \frac{3}{2} \). Find the original fraction.
Answer:
Let the numerator of the fraction be \( x \).
The denominator of the fraction is 8 more than its numerator.
So, the denominator of the fraction is \( x + 8 \).
Now, if 17 is added to the numerator, the new numerator becomes \( x + 17 \).
And if 1 is subtracted from the denominator, the new denominator becomes \( x + 8 - 1 \), which is \( x + 7 \).
Thus, the new fraction is \( \frac{x+17}{x+7} \).
According to the problem, the new fraction is equal to \( \frac{3}{2} \).
\( \frac{x+17}{x+7} = \frac {3}{2} \)
We perform cross-multiplication.
\( 2(x + 17) = 3(x + 7) \)
\( 2x + 34 = 3x + 21 \)
Next, we move \( 3x \) to the left side and \( 34 \) to the right side.
\( 2x - 3x = 21 - 34 \)
\( -x = -13 \)
We multiply both sides by \( -1 \).
\( x = 13 \)
Therefore, the numerator of the fraction is \( x = 13 \).
And the denominator is \( x + 8 = 13 + 8 = 21 \).
The fraction is \( \frac {13}{21} \).
Thus, the original fraction is \( \frac {13}{21} \).
In simple words: We first set the numerator as 'x' and the denominator as 'x + 8'. Then, we changed the numerator and denominator based on the problem's rules to get a new fraction. We set this new fraction equal to 3/2 and solved the equation to find 'x'. Finally, we used 'x' to figure out the original fraction.

Exam Tip: Clearly define your variables for the numerator and denominator before setting up the equation, and carefully apply all conditions given in the problem statement.