💡To solve for ( x ) in the equation ( frac{x - 1}{x - 5} = 0 ), I start by recognizing that a fraction equals zero when its numerator is zero, provided the denominator is not zero. First, I set the numerator equal to zero:[x - 1 = 0]Solving for ( x ), I find:[x = 1]Next, I check the denominator to ensure it is not zero when ( x = 1 ):[x - 5 = 1 - 5 = -4 neq 0]Since the denominator is not zero, ( x = 1 ) is a valid solution.

💡Okay, so I have this problem about a train crossing a bridge. Let me try to understand what's being asked here. The train is 170 meters long and is traveling at a speed of 45 kilometers per hour. It needs to cross a bridge that's 205 meters long. I need to find out how many seconds it takes for the train to completely cross the bridge.Alright, first things first, I think I need to figure out the total distance the train has to cover to completely cross the bridge. Since the train itself is 170 meters long and the bridge is 205 meters long, the total distance should be the sum of these two lengths, right? So, that would be 170 meters plus 205 meters. Let me add those up:170 meters + 205 meters = 375 metersOkay, so the train needs to travel 375 meters to completely cross the bridge. Now, the train is moving at a speed of 45 kilometers per hour. Hmm, the speed is given in kilometers per hour, but the distance is in meters. I think I need to convert the speed to meters per second to make the units consistent. That way, I can easily find the time in seconds.To convert kilometers per hour to meters per second, I remember that 1 kilometer is 1000 meters and 1 hour is 3600 seconds. So, the conversion factor should be:1 kilometer per hour = 1000 meters / 3600 secondsLet me apply that to 45 kilometers per hour:45 km/hr = 45 * (1000 meters / 3600 seconds)Let me calculate that:45 * 1000 = 45,00045,000 / 3600 = 12.5So, 45 kilometers per hour is equal to 12.5 meters per second.Now that I have the speed in meters per second and the total distance in meters, I can find the time it takes for the train to cross the bridge. Time is equal to distance divided by speed, right?Time = Distance / SpeedTime = 375 meters / 12.5 meters per secondLet me do that division:375 / 12.5 = 30So, the time it takes for the train to cross the bridge is 30 seconds.Wait, let me double-check my calculations to make sure I didn't make any mistakes. Starting with the total distance: 170 meters + 205 meters is indeed 375 meters. Then, converting 45 km/hr to m/s: 45 * 1000 / 3600 equals 12.5 m/s. That seems correct. Finally, dividing 375 by 12.5 gives 30 seconds. Yeah, that all adds up.I think I've got it right. The train takes 30 seconds to cross the bridge.

💡Okay, so I need to find the projection of the vector (begin{pmatrix} -3 6 2 end{pmatrix}) onto a line given by the vector equation (frac{x}{4} = frac{y}{-1} = frac{z}{3}). Hmm, I remember that projecting a vector onto a line involves using the direction vector of the line. First, I should figure out the direction vector of the line. The equation (frac{x}{4} = frac{y}{-1} = frac{z}{3}) suggests that the direction vector is (begin{pmatrix} 4 -1 3 end{pmatrix}). That makes sense because the denominators in the parametric equations give the direction ratios.Now, to project a vector onto another vector, I think the formula is something like the dot product of the two vectors divided by the magnitude squared of the direction vector, and then multiplied by the direction vector itself. Let me write that down:The projection of vector (mathbf{v}) onto vector (mathbf{u}) is given by:[text{proj}_{mathbf{u}} mathbf{v} = left( frac{mathbf{v} cdot mathbf{u}}{mathbf{u} cdot mathbf{u}} right) mathbf{u}]So, in this case, (mathbf{v}) is (begin{pmatrix} -3 6 2 end{pmatrix}) and (mathbf{u}) is (begin{pmatrix} 4 -1 3 end{pmatrix}).First, I need to compute the dot product (mathbf{v} cdot mathbf{u}). Let me do that step by step:[(-3) times 4 = -12][6 times (-1) = -6][2 times 3 = 6]Adding these up: (-12 + (-6) + 6 = -12). Okay, so the dot product is (-12).Next, I need the magnitude squared of (mathbf{u}). That's (mathbf{u} cdot mathbf{u}):[4^2 + (-1)^2 + 3^2 = 16 + 1 + 9 = 26]So, the magnitude squared is 26.Now, the scalar multiplier for the projection is (frac{-12}{26}), which simplifies to (-frac{6}{13}).Finally, I multiply this scalar by the direction vector (mathbf{u}) to get the projection vector:[-frac{6}{13} times begin{pmatrix} 4 -1 3 end{pmatrix} = begin{pmatrix} -frac{24}{13} frac{6}{13} -frac{18}{13} end{pmatrix}]Wait, let me double-check my calculations to make sure I didn't make any mistakes. The dot product was (-12), and the magnitude squared was 26, so the scalar is indeed (-frac{6}{13}). Multiplying each component of the direction vector by this scalar gives the projection vector as above. Hmm, does this make sense? The projection vector should lie along the direction of the line, which it does since it's a scalar multiple of the direction vector. The negative scalar just means it's pointing in the opposite direction of the direction vector. I think that's correct. So, the projection of (begin{pmatrix} -3 6 2 end{pmatrix}) onto the line is (begin{pmatrix} -frac{24}{13} frac{6}{13} -frac{18}{13} end{pmatrix}).

💡First, calculate the total number of days Al needs to take the pills. Since three weeks equal 21 days, and he takes one green pill and one pink pill each day, the total number of pills is 42.Let ( g ) represent the cost of one green pill and ( p ) represent the cost of one pink pill. According to the problem, ( g = p + 1 ).The total cost for all the pills is 819. This can be expressed as:[ 21g + 21p = 819 ]Substituting ( g ) with ( p + 1 ):[ 21(p + 1) + 21p = 819 ]Simplify the equation:[ 21p + 21 + 21p = 819 ][ 42p + 21 = 819 ]Subtract 21 from both sides:[ 42p = 798 ]Divide both sides by 42:[ p = 19 ]Now, find ( g ):[ g = p + 1 = 19 + 1 = 20 ]Therefore, the cost of one green pill is 20.

💡I need to simplify the fraction (frac{48}{72}).First, I'll find the greatest common divisor (GCD) of 48 and 72 to determine the largest number that divides both without a remainder.I can list the factors of each number:- Factors of 48: 1, 2, 3, 4, 6, 8, 12, 16, 24, 48- Factors of 72: 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72The common factors are 1, 2, 3, 4, 6, 8, 12, and 24. The greatest common factor is 24.Next, I'll divide both the numerator and the denominator by the GCD (24):[frac{48 div 24}{72 div 24} = frac{2}{3}]So, the simplified form of (frac{48}{72}) is (frac{2}{3}).

💡Okay, so I have this quadratic expression: 3x² + 9x + 20, and I need to rewrite it in the form a(x - h)² + k. Hmm, I think this is called completing the square. I remember that completing the square helps in converting a quadratic from standard form to vertex form, which is what this question is asking for.First, let me recall the general steps for completing the square. I think it involves factoring out the coefficient of x² from the first two terms, then creating a perfect square trinomial inside the parentheses. After that, I adjust the constants to maintain the equality of the expression.So, starting with 3x² + 9x + 20. The coefficient of x² is 3, so I should factor that out from the first two terms. Let me try that:3(x² + 3x) + 20.Okay, that looks right. Now, inside the parentheses, I have x² + 3x. To complete the square, I need to find a number that, when added and subtracted, will make this a perfect square trinomial. The formula I remember is taking half of the coefficient of x, squaring it, and then adding it inside the parentheses. But since I've factored out a 3, I need to be careful with how adding and subtracting affects the overall expression.So, the coefficient of x is 3. Half of 3 is 1.5, which is 3/2. Squaring that gives (3/2)² = 9/4. So, I need to add and subtract 9/4 inside the parentheses. Let me write that down:3(x² + 3x + 9/4 - 9/4) + 20.Wait, actually, since I added 9/4 inside the parentheses, which is multiplied by 3, I need to subtract the same amount multiplied by 3 to keep the expression balanced. Let me correct that:3(x² + 3x + 9/4) - 3*(9/4) + 20.Simplifying that, the expression becomes:3(x + 3/2)² - 27/4 + 20.Now, I need to combine the constant terms: -27/4 and 20. To do that, I'll convert 20 into fourths. 20 is equal to 80/4, so:-27/4 + 80/4 = (80 - 27)/4 = 53/4.So, putting it all together, the expression is:3(x + 3/2)² + 53/4.But the question asks for the form a(x - h)² + k. Right now, I have (x + 3/2)², which can be written as (x - (-3/2))². So, h is -3/2.Wait, let me double-check my steps to make sure I didn't make a mistake. Factoring out the 3, completing the square by adding and subtracting 9/4, adjusting for the factored 3, and then combining constants. Yeah, that seems correct. So, h should indeed be -3/2.