Verified Answer :
- Isolate variables by performing inverse operations (division for multiplication, multiplication for division).
- Verify each solution by substituting the result back into the original equation.
- Results: \(h=5, t=9, n=-14, m=-2, t=-4, p=-8, d=-9, n=8, g=-16, c=-20\).
- Final solutions: \(\boxed{h=5, t=9, n=-14, m=-2, t=-4, p=-8, d=-9, n=8, g=-16, c=-20}\).
Explanation
- Understanding the Problem and Strategy
We are presented with a series of linear equations that need to be solved for their respective variables. A linear equation in one variable typically takes the form \(ax = b\) or \(\frac{x}{a} = b\). To solve an equation of the form \(ax = b\), we isolate the variable \(x\) by dividing both sides of the equation by the coefficient \(a\). To solve an equation of the form \(\frac{x}{a} = b\), we isolate the variable \(x\) by multiplying both sides of the equation by the denominator \(a\). After finding the solution, we can verify it by substituting the value back into the original equation to ensure the equality holds. We will address each of the complete equations provided in the list. Note that equations 13 and 21 are provided without a value for their right-hand side, so they cannot be solved as given.
- Solving the Equations
We solve the equations one by one:
- \(3h = 15\): Divide both sides by 3: \(h = \frac{15}{3} = 5\). Check: \(3(5) = 15\). Correct.
- \(-5t = -45\): Divide both sides by -5: \(t = \frac{-45}{-5} = 9\). Check: \(-5(9) = -45\). Correct.
- \(\frac{n}{2} = -7\): Multiply both sides by 2: \(n = -7 \times 2 = -14\). Check: \(\frac{-14}{2} = -7\). Correct.
- \(5m = -10\): Divide both sides by 5: \(m = \frac{-10}{5} = -2\). Check: \(5(-2) = -10\). Correct.
- \(8t = -32\): Divide both sides by 8: \(t = \frac{-32}{8} = -4\). Check: \(8(-4) = -32\). Correct.
- \(-6p = 48\): Divide both sides by -6: \(p = \frac{48}{-6} = -8\). Check: \(-6(-8) = 48\). Correct.
- \(-72 = 8d\): Divide both sides by 8: \(d = \frac{-72}{8} = -9\). Check: \(8(-9) = -72\). Correct.
- \(\frac{n}{1.6} = 5\): Multiply both sides by 1.6: \(n = 5 \times 1.6 = 8\). Check: \(\frac{8}{1.6} = 5\). Correct.
- \(\frac{3}{4}g = -12\): Multiply both sides by \(\frac{4}{3}\): \(g = -12 \times \frac{4}{3} = -4 \times 4 = -16\). Check: \(\frac{3}{4}(-16) = -12\). Correct.
- \(8 = -\frac{2}{5}c\): Multiply both sides by \(-\frac{5}{2}\): \(c = 8 \times (-\frac{5}{2}) = 4 \times (-5) = -20\). Check: \(-\frac{2}{5}(-20) = 8\). Correct.
- Conclusion and Final Results
The solutions for the equations are as follows:
- \(h = 5\)
- \(t = 9\)
- \(n = -14\)
- \(m = -2\)
- \(t = -4\)
- \(p = -8\)
- \(d = -9\)
- \(n = 8\)
- \(g = -16\)
- \(c = -20\)
Equations 13 and 21 could not be solved as the right-hand side was not provided. We have verified each solution by substituting the value back into the respective original equations.
Examples
Linear equations are foundational in real-life applications involving rates and quantities. For instance, if you are buying items at a fixed price, like purchasing several notebooks for a total cost, you can represent this as a linear equation: \(price \times quantity = total\_cost\). By solving for the unknown variable, you can determine either the price per item or the number of items you can afford. Mastering these simple algebraic manipulations is the first step in modeling more complex scenarios such as budgeting, construction planning, and data analysis.