Answered

Equations with Variables on Both Sides



1. $-2 a-9=4 a=15$		2. $36 + 3n - 5n - 6$

3. $7 x+12=x-6$		4.

5. $\frac{1}{2} x-5=10-\frac{3}{4} x$		6. $\frac{2}{3} y+1=\frac{1}{6} y+8$

7. $5.4 p+13.1=-2.6 p+3.5$		8. $0.3 w+0.35=0.22 w-0$.

Verified Answer :

  • Simplify item 2 to \(30 - 2n\).
  • Equation 1 \(-2a - 9 = 4a - 15\) results in \(a = 1\).
  • Equation 3 \(7x + 12 = x - 6\) results in \(x = -3\).
  • Equation 5 \(\frac{1}{2}x - 5 = 10 - \frac{3}{4}x\) results in \(x = 12\).
  • Equation 6 \(\frac{2}{3}y + 1 = \frac{1}{6}y + 8\) results in \(y = 14\).
  • Equation 7 \(5.4p + 13.1 = -2.6p + 3.5\) results in \(p = -1.2\).
  • Equation 8 \(0.3w + 0.35 = 0.22w - 0.4\) results in \(w = -9.375\).
  • Final results: \(\boxed{a=1, n=30-2n, x_3=-3, x_5=12, y=14, p=-1.2, w=-9.375}\)

Explanation

  1. Problem Analysis and Strategy

The goal is to solve several linear equations for their respective variables. A linear equation involves variables and constants, and the strategy is to isolate the variable by performing the same operations on both sides of the equation. We will treat each equation one by one:

  1. \(-2a - 9 = 4a - 15\)
  2. \(36 + 3n - 5n - 6\) (Expression)
  3. \(7x + 12 = x - 6\)
  4. (Illegible)
  5. \(\frac{1}{2}x - 5 = 10 - \frac{3}{4}x\)
  6. \(\frac{2}{3}y + 1 = \frac{1}{6}y + 8\)
  7. \(5.4p + 13.1 = -2.6p + 3.5\)
  8. \(0.3w + 0.35 = 0.22w - 0.4\)

Note that for item 2, we have an expression, not an equation, so we will simplify it instead. For item 4, we exclude it from the resolution as it is illegible.

  1. Solving First Three Items

For the first equation \(-2a - 9 = 4a - 15\):

  • Add \(2a\) to both sides: \(-9 = 6a - 15\)
  • Add \(15\) to both sides: \(6 = 6a\)
  • Divide by \(6\): \(a = 1\)

For the expression \(36 + 3n - 5n - 6\):

  • Combine like terms: \((36 - 6) + (3n - 5n) = 30 - 2n\)

For the third equation \(7x + 12 = x - 6\):

  • Subtract \(x\) from both sides: \(6x + 12 = -6\)
  • Subtract \(12\) from both sides: \(6x = -18\)
  • Divide by \(6\): \(x = -3\)
  1. Solving Equations with Fractions

For the fifth equation \(\frac{1}{2}x - 5 = 10 - \frac{3}{4}x\):

  • Multiply the entire equation by \(4\) to clear denominators: \(2x - 20 = 40 - 3x\)
  • Add \(3x\) to both sides: \(5x - 20 = 40\)
  • Add \(20\) to both sides: \(5x = 60\)
  • Divide by \(5\): \(x = 12\)

For the sixth equation \(\frac{2}{3}y + 1 = \frac{1}{6}y + 8\):

  • Multiply the entire equation by \(6\): \(4y + 6 = y + 48\)
  • Subtract \(y\) from both sides: \(3y + 6 = 48\)
  • Subtract \(6\) from both sides: \(3y = 42\)
  • Divide by \(3\): \(y = 14\)
  1. Solving Equations with Decimals and Conclusion

For the seventh equation \(5.4p + 13.1 = -2.6p + 3.5\):

  • Add \(2.6p\) to both sides: \(8p + 13.1 = 3.5\)
  • Subtract \(13.1\) from both sides: \(8p = -9.6\)
  • Divide by \(8\): \(p = -1.2\)

For the eighth equation \(0.3w + 0.35 = 0.22w - 0.4\):

  • Subtract \(0.22w\) from both sides: \(0.08w + 0.35 = -0.4\)
  • Subtract \(0.35\) from both sides: \(0.08w = -0.75\)
  • Divide by \(0.08\): \(w = -9.375\)

The solutions to the equations are:
\(a = 1\), \(x = -3\) (from eq 3), \(x = 12\) (from eq 5), \(y = 14\), \(p = -1.2\), \(w = -9.375\). The expression (item 2) simplifies to \(30 - 2n\).

Examples

Linear equations are fundamental for modeling real-world situations involving constant rates of change. For example, consider two different cell phone plans: Plan A charges a flat fee of \(\$30\) plus \(\$2\) per GB of data, and Plan B charges a flat fee of \(\$6\) plus \(\$3\) per GB. To find the amount of data usage where both plans cost the same, you can set up the equation \(30 + 2g = 6 + 3g\). Solving this for \(g\) gives \(g = 24\) GB, indicating that at 24 GB of usage, both plans will cost the same amount, helping the consumer make an informed decision based on their expected data consumption.

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