This question tests your ability to rearrange formulas to solve for a specific variable—essential for using formulas flexibly in science, engineering, and real-world problem-solving. More complex rearrangements may require advanced techniques: if your target variable is squared (like $r^2$ in $A = \pi r^2$), you'll need square roots ($r = \sqrt{\frac{A}{\pi}}$). If it appears in a denominator (like $f$ in $\frac{1}{f} = \frac{1}{a} + \frac{1}{b}$), you'll need to clear fractions first. If it appears with different powers (like $t$ in $s = ut + \frac{1}{2}at^2$), you may need the quadratic formula! The complexity of the rearrangement depends on how the variable appears in the formula. Starting with $s = ut + \frac{1}{2}at^2$, we need to solve for $t$. First multiply through by 2: $2s = 2ut + at^2$. Rearranging: $at^2 + 2ut - 2s = 0$. This is a quadratic in standard form with $A = a$, $B = 2u$, $C = -2s$. Using the quadratic formula: $t = \frac{-2u \pm \sqrt{4u^2 + 8as}}{2a} = \frac{-2u \pm 2\sqrt{u^2 + 2as}}{2a} = \frac{-u \pm \sqrt{u^2 + 2as}}{a}$. Choice A correctly applies the quadratic formula to get $t = \frac{-u \pm \sqrt{u^2 + 2as}}{a}$. Choice B has the wrong sign on u, Choice C has the wrong sign under the square root (should be +2as not -2as), and Choice D incorrectly divides by 2a instead of a. The formula rearrangement recipe: (1) Identify what you're solving for (that's your 'x'), (2) Identify what operation(s) are being done to that variable in the original formula, (3) Apply inverse operations in reverse order to isolate it (just like numeric equations!), (4) Simplify the result—combine fractions, simplify radicals, etc.

This question tests your ability to rearrange formulas to solve for a specific variable—essential for using formulas flexibly in science, engineering, and real-world problem-solving. More complex rearrangements may require advanced techniques: if your target variable is squared (like r² in A = πr²), you'll need square roots (r = √(A/π)). If it appears in a denominator (like f in 1/f = 1/a + 1/b), you'll need to clear fractions first. If it appears with different powers (like t in s = ut + (1/2)at²), you may need the quadratic formula! The complexity of the rearrangement depends on how the variable appears in the formula. Starting with V = (1/3)πr²h, we need to isolate r. First multiply both sides by 3: 3V = πr²h. Then divide both sides by πh: 3V/(πh) = r². Finally, take the square root: r = √[3V/(πh)]. Choice B correctly isolates r through multiplication, division, and taking the square root to get r = √(3V/(πh)). Choice A forgets the factor of 3, Choice C forgets to take the square root (leaving r²), and Choice D incorrectly inverts the expression under the square root. The formula rearrangement recipe: (1) Identify what you're solving for (that's your 'x'), (2) Identify what operation(s) are being done to that variable in the original formula, (3) Apply inverse operations in reverse order to isolate it (just like numeric equations!), (4) Simplify the result—combine fractions, simplify radicals, etc.

This question tests your ability to rearrange formulas to solve for a specific variable—essential for using formulas flexibly in science, engineering, and real-world problem-solving. Watch for variables appearing multiple times: if your target variable appears in multiple places (like x in y = kx/(x + a)), factor it out first. Starting with y = kx/(x + a), multiply both sides by (x + a) to get y(x + a) = kx, expand to yx + ya = kx, then bring terms with x to one side: ya = kx - yx = x(k - y), so x = ya/(k - y). Choice A correctly isolates x through multiplication, rearrangement, and factoring to get x = ya/(k - y). A distractor like choice B might flip the denominator sign, but subtracting y from k keeps it positive assuming k > y. The formula rearrangement recipe: (1) Identify what you're solving for (that's your 'x'), (2) Identify what operation(s) are being done to that variable in the original formula, (3) Apply inverse operations in reverse order to isolate it (just like numeric equations!), (4) Simplify the result—combine fractions, simplify radicals, etc. You're mastering algebraic models—fantastic progress!

This question tests your ability to rearrange formulas to solve for a specific variable—essential for using formulas flexibly in science, engineering, and real-world problem-solving. Rearranging formulas with multiple variables works exactly like solving numeric equations, except the answer contains other variables instead of numbers: treat the variable you're solving for as the 'unknown x,' treat all other variables as 'known numbers,' then use inverse operations to isolate your target variable. Starting with PV = nRT, divide both sides by nR to isolate T, resulting in T = PV / (nR). Choice A correctly isolates T through division to get T = PV / (nR). A distractor like choice D might misplace the n in the numerator, but remember to divide by all factors multiplying T. The formula rearrangement recipe: (1) Identify what you're solving for (that's your 'x'), (2) Identify what operation(s) are being done to that variable in the original formula, (3) Apply inverse operations in reverse order to isolate it (just like numeric equations!), (4) Simplify the result—combine fractions, simplify radicals, etc. Excellent work with gas laws—you've got this!

This question tests your ability to rearrange formulas to solve for a specific variable—essential for using formulas flexibly in science, engineering, and real-world problem-solving. Rearranging formulas with multiple variables works exactly like solving numeric equations, except the answer contains other variables instead of numbers: treat the variable you're solving for as the 'unknown x,' treat all other variables as 'known numbers,' then use inverse operations to isolate your target variable. The same algebraic moves apply—just keep everything symbolic! For example, solving V = πr²h for r is like solving 100 = 3.14·r²·5 for r: divide by π and h, then take square root. Starting with v² = u² + 2as, we need to isolate a. First subtract u² from both sides: v² - u² = 2as. Then divide both sides by 2s: (v² - u²)/(2s) = a, which gives us a = (v² - u²)/(2s). Choice B correctly isolates a through subtraction and division to get a = (v² - u²)/(2s). Choice A forgets to divide by 2, Choice C incorrectly switches the order in the numerator (should be v² - u², not u² - v²), and Choice D inverts the entire fraction. The formula rearrangement recipe: (1) Identify what you're solving for (that's your 'x'), (2) Identify what operation(s) are being done to that variable in the original formula, (3) Apply inverse operations in reverse order to isolate it (just like numeric equations!), (4) Simplify the result—combine fractions, simplify radicals, etc.

This question tests your ability to rearrange formulas to solve for a specific variable—essential for using formulas flexibly in science, engineering, and real-world problem-solving. Rearranging formulas with multiple variables works exactly like solving numeric equations, except the answer contains other variables instead of numbers: treat the variable you're solving for as the 'unknown x,' treat all other variables as 'known numbers,' then use inverse operations to isolate your target variable. Starting with $v^2 = u^2 + 2as$, subtract $u^2$ from both sides to get $v^2 - u^2 = 2as$, then divide both sides by $2s$ to isolate a, resulting in $a = \dfrac{v^2 - u^2}{2s}$. Choice B correctly isolates a through subtraction and division to get $a = \dfrac{v^2 - u^2}{2s}$. A common distractor like choice C might forget to switch the signs when subtracting $u^2$, but remember that $v^2 - u^2$ is positive if v > u, so the order matters for the physics context. The formula rearrangement recipe: (1) Identify what you're solving for (that's your 'x'), (2) Identify what operation(s) are being done to that variable in the original formula, (3) Apply inverse operations in reverse order to isolate it (just like numeric equations!), (4) Simplify the result—combine fractions, simplify radicals, etc. Keep practicing these, and you'll master kinematic equations in no time!

This question tests your ability to rearrange formulas to solve for a specific variable—essential for using formulas flexibly in science, engineering, and real-world problem-solving. More complex rearrangements may require advanced techniques: if your target variable is squared (like r² in A = πr²), you'll need square roots (r = √(A/π)). If it appears in a denominator (like f in 1/f = 1/a + 1/b), you'll need to clear fractions first. If it appears with different powers (like t in s = ut + (1/2)at²), you may need the quadratic formula! The complexity of the rearrangement depends on how the variable appears in the formula. Starting with KE = (1/2)mv², we need to isolate v. First multiply both sides by 2: 2KE = mv². Then divide both sides by m: 2KE/m = v². Finally, take the square root: v = ±√(2KE/m). The ± is crucial because squaring either positive or negative v gives the same v². Choice B correctly isolates v through multiplication, division, and taking the square root to get v = ±√(2KE/m). Choice A forgets the square root, Choice C has the wrong coefficient (should be 2KE not KE), and Choice D incorrectly inverts the expression under the square root. Watch for variables appearing multiple times: if your target variable appears in multiple places (like x in xy + xz = w), factor it out first: x(y + z) = w, then x = w/(y + z). If you don't factor, you'll struggle to isolate! Also, when taking square roots of a variable, remember ± if the formula context allows both positive and negative (though often context restricts to positive only, like radius r ≥ 0). Physics and geometry formulas usually want positive values only!

This question tests your ability to rearrange formulas to solve for a specific variable—essential for using formulas flexibly in science, engineering, and real-world problem-solving. Rearranging formulas with multiple variables works exactly like solving numeric equations, except the answer contains other variables instead of numbers: treat the variable you're solving for as the 'unknown x,' treat all other variables as 'known numbers,' then use inverse operations to isolate your target variable. The same algebraic moves apply—just keep everything symbolic! For example, solving $V = \pi r^2 h$ for r is like solving $100 = 3.14 \cdot r^2 \cdot 5$ for r: divide by $\pi$ and $h$, then take square root. Starting with $P = \dfrac{nRT}{V}$, we need to isolate T. First multiply both sides by V: $PV = nRT$. Then divide both sides by nR: $\dfrac{PV}{nR} = T$, which gives us $T = \dfrac{PV}{nR}$. Choice A correctly isolates T through multiplication and division to get $T = \dfrac{PV}{nR}$. Choice B inverts the expression, Choice C has only P in the numerator instead of PV, and Choice D incorrectly has nRP in the numerator. Watch for variables appearing multiple times: if your target variable appears in multiple places (like x in $xy + xz = w$), factor it out first: $x(y + z) = w$, then $x = \dfrac{w}{y + z}$. If you don't factor, you'll struggle to isolate! Also, when taking square roots of a variable, remember $\pm$ if the formula context allows both positive and negative (though often context restricts to positive only, like radius $r \geq 0$). Physics and geometry formulas usually want positive values only!

This question tests your ability to rearrange formulas to solve for a specific variable—essential for using formulas flexibly in science, engineering, and real-world problem-solving. More complex rearrangements may require advanced techniques: if your target variable is squared (like r² in A = πr²), you'll need square roots (r = √(A/π)). If it appears in a denominator (like f in 1/f = 1/a + 1/b), you'll need to clear fractions first. If it appears with different powers (like t in s = ut + (1/2)at²), you may need the quadratic formula! The complexity of the rearrangement depends on how the variable appears in the formula. Starting with A = πr², we need to isolate r. First divide both sides by π: A/π = r². Then take the square root of both sides: √(A/π) = r, which gives us r = √(A/π). Choice A correctly isolates r through division by π and taking the square root to get r = √(A/π). Choice B incorrectly divides by π² instead of π, Choice C forgets to take the square root (leaving r²), and Choice D incorrectly multiplies inside the square root instead of dividing. Watch for variables appearing multiple times: if your target variable appears in multiple places (like x in xy + xz = w), factor it out first: x(y + z) = w, then x = w/(y + z). If you don't factor, you'll struggle to isolate! Also, when taking square roots of a variable, remember ± if the formula context allows both positive and negative (though often context restricts to positive only, like radius r ≥ 0). Physics and geometry formulas usually want positive values only!

This question tests your ability to rearrange formulas to solve for a specific variable—essential for using formulas flexibly in science, engineering, and real-world problem-solving. Rearranging formulas with multiple variables works exactly like solving numeric equations, except the answer contains other variables instead of numbers: treat the variable you're solving for as the 'unknown x,' treat all other variables as 'known numbers,' then use inverse operations to isolate your target variable. The same algebraic moves apply—just keep everything symbolic! For example, solving $V = \pi r^2 h$ for r is like solving $100 = 3.14 \cdot r^2 \cdot 5$ for r: divide by $\pi$ and h, then take square root. Starting with $y = \dfrac{x-a}{b} + c$, we need to isolate x. First subtract c from both sides: $y - c = \dfrac{x-a}{b}$. Then multiply both sides by b: $b(y - c) = x - a$. Finally, add a to both sides: $b(y - c) + a = x$, which gives us $x = b(y - c) + a$. Choice B correctly isolates x through subtraction, multiplication, and addition to get $x = b(y - c) + a$. Choice A incorrectly expands to by - a + c, Choice C divides instead of multiplying by b, and Choice D has the wrong sign on c inside the parentheses. The formula rearrangement recipe: (1) Identify what you're solving for (that's your 'x'), (2) Identify what operation(s) are being done to that variable in the original formula, (3) Apply inverse operations in reverse order to isolate it (just like numeric equations!), (4) Simplify the result—combine fractions, simplify radicals, etc.