solution

-28 is the solution to the sum of a number p and 21 The sum of a number p and 21: p + 21 The phrase [I]is the solution to[/I] means an equation, so we set p + 21 equal to -28: [B]p + 21 = -28 [/B] If the problem asks you to solve for p, then we [URL='https://www.mathcelebrity.com/1unk.php?num=p%2B21%3D-28&pl=Solve']type this into our search engine[/URL] and we get: p = [B]-49[/B]

2 numbers add to 200. The first is 20 less than the second. Let the first number be x and the second number be y. We're given: [LIST=1] [*]x + y = 200 [*]x = y - 20 [/LIST] Plug (2) into (1) (y - 20) + y = 200 Group like terms: 2y - 20 = 200 [URL='https://www.mathcelebrity.com/1unk.php?num=2y-20%3D200&pl=Solve']Typing this equation into our search engine[/URL], we get: [B]y = 110[/B] <-- This is the larger number Plug y = 110 into Equation (2) to get the smaller number: x = 110 - 20 [B]x = 90[/B] <-- This is the smaller number Let's check our work for Equation (1) using x = 90, and y = 110 90 + 110 ? 200 200 = 200 <-- Good, our solutions check out for equation (1) Let's check our work for Equation (2) using x = 90, and y = 110 90 = 110 - 20 90 = 90 <-- Good, our solutions check out for equation (2)

A 12% acid solution is made by mixing 8% and 20% solutions. If the 450 ml of the 12% solution is required, how much of each solution is required? Component Unit Amount 8% Solution: 0.08 * x = 0.08x 20% Solution: 0.2 * y = 0.2y 12% Solution: 0.12 * 450 = 54 We add up the 8% solution and 20% solution to get two equations: [LIST=1] [*]0.08x + 0.2y = 54 [*]x + y = 450 [/LIST] We have a simultaneous set of equations. We can solve it using three methods: [LIST] [*][URL='https://www.mathcelebrity.com/simultaneous-equations.php?term1=0.08x+%2B+0.2y+%3D+54&term2=x+%2B+y+%3D+450&pl=Substitution']Substitution Method[/URL] [*][URL='https://www.mathcelebrity.com/simultaneous-equations.php?term1=0.08x+%2B+0.2y+%3D+54&term2=x+%2B+y+%3D+450&pl=Elimination']Elimination Method[/URL] [*][URL='https://www.mathcelebrity.com/simultaneous-equations.php?term1=0.08x+%2B+0.2y+%3D+54&term2=x+%2B+y+%3D+450&pl=Cramers+Method']Cramer's Rule[/URL] [/LIST] No matter which method we choose, we get the same answer: [LIST] [*][B]x = 300 ml[/B] [*][B]y = 150 ml[/B] [/LIST]

A 3 hour river cruise goes 15 km upstream and then back again. The river has a current of 2 km an hour. What is the boat's speed and how long was the upstream journey? [U]Set up the relationship of still water speed and downstream speed[/U] Speed down stream = Speed in still water + speed of the current Speed down stream = x+2 Therefore: Speed upstream =x - 2 Since distance = rate * time, we rearrange to get time = Distance/rate: 15/(x+ 2) + 15 /(x- 2) = 3 Multiply each side by 1/3 and we get: 5/(x + 2) + 5/(x - 2) = 1 Using a common denominator of (x + 2)(x - 2), we get: 5(x - 2)/(x + 2)(x - 2) + 5(x + 2)/(x + 2)(x - 2) (5x - 10 + 5x + 10)/5(x - 2)/(x + 2)(x - 2) 10x = (x+2)(x-2) We multiply through on the right side to get: 10x = x^2 - 4 Subtract 10x from each side: x^2 - 10x - 4 = 0 This is a quadratic equation. To solve it, [URL='https://www.mathcelebrity.com/quadratic.php?num=x%5E2-10x-4%3D0&pl=Solve+Quadratic+Equation&hintnum=+0']we type it in our search engine[/URL] and we get: Speed of the boat in still water =X=5 +- sq. Root of 29 kmph We only want the positive solution: x = 5 + sqrt(29) x = 10.38 [U]Calculate time for upstream journey:[/U] Time for upstream journey = 15/(10.38 - 2) Time for upstream journey = 15/(8.38) Time for upstream journey = [B]1.79[/B] [U]Calculate time for downstream journey:[/U] Time for downstream journey = 15/(10.38 + 2) Time for downstream journey = 15/(12.38) Time for downstream journey = [B]1.21[/B]

Let x be the number. We have: x^2 + x = 30 Subtract 30 from each side: x^2 + x - 30 = 0 Using our [URL='http://www.mathcelebrity.com/quadratic.php?num=x%5E2%2Bx-30%3D0&pl=Solve+Quadratic+Equation&hintnum=+0']quadratic calculator[/URL], we get potential solutions of: [B]x = 5 or x = -6[/B] Check 5: 5 + 5^2 = 5 + 25 = [B]30[/B] Check -6 -6 + -6^2 = -6 + 36 = [B]30[/B]

A checking account is set up with an initial balance of $2400 and $200 are removed from the account each month for rent right and equation who solution is the number of months and it takes for the account balance to reach 1000 200 is removed, so we subtract. Let m be the number of months. We want the following equation: [B]2400 - 200m = 1000 [/B] Now, we want to solve this equation for m. So [URL='https://www.mathcelebrity.com/1unk.php?num=2400-200m%3D1000&pl=Solve']we type it in our search engine[/URL] and we get: m = [B]7[/B]

A farmer is taking her eggs to the market in a cart, but she hits a pothole, which knocks over all the containers of eggs. Though she is unhurt, every egg is broken. So she goes to her insurance agent, who asks her how many eggs she had. She says she doesn't know, but she remembers somethings from various ways she tried packing the eggs. When she put the eggs in groups of two, three, four, five, and six there was one egg left over, but when she put them in groups of seven they ended up in complete groups with no eggs left over. What can the farmer figure from this information about the number of eggs she had? Is there more than one answer? We need a number (n) that leaves a remainder of 1 when divided by 2, 3, 4, 5, 6 but no remainder when divided by 7. 217 + 84 = [B]301[/B]. Other solutions are multiples of 3 x 4 x 5 x 7, but we want the lowest one here.

A first number plus twice a second number is 11. Twice the first number plus the second totals 34. Find the numbers. Let the first number be x and the second number be y. We're given: [LIST=1] [*]x + 2y = 11 [*]2x + y = 34 [/LIST] Using our simultaneous equations calculator, we have 3 methods to solve this: [LIST=1] [*][URL='https://www.mathcelebrity.com/simultaneous-equations.php?term1=x+%2B+2y+%3D+11&term2=2x+%2B+y+%3D+34&pl=Substitution']Substitution Method[/URL] [*][URL='https://www.mathcelebrity.com/simultaneous-equations.php?term1=x+%2B+2y+%3D+11&term2=2x+%2B+y+%3D+34&pl=Elimination']Elimination Method[/URL] [*][URL='https://www.mathcelebrity.com/simultaneous-equations.php?term1=x+%2B+2y+%3D+11&term2=2x+%2B+y+%3D+34&pl=Cramers+Method']Cramer's Rule[/URL] [/LIST] All 3 methods give the same solution: [LIST] [*][B]x = 19[/B] [*][B]y = -4[/B] [/LIST]

At the end of the 2021 NBA season, the NY Knicks had 10 more wins than losses. This NBA season the NY Knicks played a total of 72 times. Find a solution to this problem and explain. Let w be the number of wins Let l be the number of losses We're given two equations: [LIST=1] [*]w = l + 10 [*]l + w = 72 [/LIST] To solve this system of equations, substitute equation (1) into equation (2) for w: l + l + 10 = 72 To solve this equation for l, we [URL='https://www.mathcelebrity.com/1unk.php?num=l%2Bl%2B10%3D72&pl=Solve']type it in our math engine[/URL] and we get: l = [B]31 [/B] To solve for w, we substitute l = 31 into equation (1): w = 31 + 10 w = [B]41[/B]

the calculator will use the Chinese Remainder Theorem to find the lowest possible solution for x in each modulus equation.
Given that the ni portions are not pairwise coprime and you entered two modulo equations, then the calculator will attempt to solve using the Method of Successive Subsitution

Diana invested $3000 in a savings account for 3 years. She earned $450 in interest over that time period. What interest rate did she earn? Use the formula I=Prt to find your answer, where I is interest, P is principal, r is rate and t is time. Enter your solution in decimal form rounded to the nearest hundredth. For example, if your solution is 12%, you would enter 0.12. Our givens are: [LIST] [*]I = 450 [*]P = 3000 [*]t = 3 [*]We want r [/LIST] 450 = 3000(r)(3) 450 = 9000r Divide each side by 9000 [B]r = 0.05[/B]

difference between 2 positive numbers is 3 and the sum of their squares is 117 Declare variables for each of the two numbers: [LIST] [*]Let the first variable be x [*]Let the second variable be y [/LIST] We're given 2 equations: [LIST=1] [*]x - y = 3 [*]x^2 + y^2 = 117 [/LIST] Rewrite equation (1) in terms of x by adding y to each side: [LIST=1] [*]x = y + 3 [*]x^2 + y^2 = 117 [/LIST] Substitute equation (1) into equation (2) for x: (y + 3)^2 + y^2 = 117 Evaluate and simplify: y^2 + 3y + 3y + 9 + y^2 = 117 Combine like terms: 2y^2 + 6y + 9 = 117 Subtract 117 from each side: 2y^2 + 6y + 9 - 117 = 117 - 117 2y^2 + 6y - 108 = 0 This is a quadratic equation: Solve the quadratic equation 2y2+6y-108 = 0 With the standard form of ax2 + bx + c, we have our a, b, and c values: a = 2, b = 6, c = -108 Solve the quadratic equation 2y^2 + 6y - 108 = 0 The quadratic formula is denoted below: y = -b sqrt(b^2 - 4ac)/2a [U]Step 1 - calculate negative b:[/U] -b = -(6) -b = -6 [U]Step 2 - calculate the discriminant ?:[/U] ? = b2 - 4ac: ? = 62 - 4 x 2 x -108 ? = 36 - -864 ? = 900 <--- Discriminant Since ? is greater than zero, we can expect two real and unequal roots. [U]Step 3 - take the square root of the discriminant ?:[/U] ?? = ?(900) ?? = 30 [U]Step 4 - find numerator 1 which is -b + the square root of the Discriminant:[/U] Numerator 1 = -b + ?? Numerator 1 = -6 + 30 Numerator 1 = 24 [U]Step 5 - find numerator 2 which is -b - the square root of the Discriminant:[/U] Numerator 2 = -b - ?? Numerator 2 = -6 - 30 Numerator 2 = -36 [U]Step 6 - calculate your denominator which is 2a:[/U] Denominator = 2 * a Denominator = 2 * 2 Denominator = 4 [U]Step 7 - you have everything you need to solve. Find solutions:[/U] Solution 1 = Numerator 1/Denominator Solution 1 = 24/4 Solution 1 = 6 Solution 2 = Numerator 2/Denominator Solution 2 = -36/4 Solution 2 = -9 [U]As a solution set, our answers would be:[/U] (Solution 1, Solution 2) = (6, -9) Since one of the solutions is not positive and the problem asks for 2 positive number, this problem has no solution

Free Diophantine Equations Calculator - Solves for ax + by = c using integer solutions if they exist

Find two consecutive integers if the sum of their squares is 1513 Let the first integer be n. The next consecutive integer is (n + 1). The sum of their squares is: n^2 + (n + 1)^2 = 1513 n^2 + n^2 + 2n + 1 = 1513 2n^2 + 2n + 1 = 1513 Subtract 1513 from each side: 2n^2 + 2n - 1512 = 0 We have a quadratic equation. We [URL='https://www.mathcelebrity.com/quadratic.php?num=2n%5E2%2B2n-1512%3D0&pl=Solve+Quadratic+Equation&hintnum=+0']type this into our search engine[/URL] and get: n = (-27, 28) Let's take the positive solution. The second integer is: n + 1 28 + 1 = 29

40d = 60 + 40d d has no solution. Are you sure it was 40 days for both Company A and B?

If 7 times the square of an integer is added to 5 times the integer, the result is 2. Find the integer. [LIST] [*]Let the integer be "x". [*]Square the integer: x^2 [*]7 times the square: 7x^2 [*]5 times the integer: 5x [*]Add them together: 7x^2 + 5x [*][I]The result is[/I] means an equation, so we set 7x^2 + 5x equal to 2 [/LIST] 7x^2 + 5x = 2 [U]This is a quadratic equation. To get it into standard form, we subtract 2 from each side:[/U] 7x^2 + 5x - 2 = 2 - 2 7x^2 + 5x - 2 = 0 [URL='https://www.mathcelebrity.com/quadratic.php?num=7x%5E2%2B5x-2%3D0&pl=Solve+Quadratic+Equation&hintnum=+0']Type this problem into our search engine[/URL], and we get two solutions: [LIST=1] [*]x = 2/7 [*]x= -1 [/LIST] The problem asks for an integer, so our answer is x[B] = -1[/B]. [U]Let's check our work by plugging x = -1 into the quadratic:[/U] 7x^2 + 5x - 2 = 0 7(-1)^2 + 5(-1) - 2 ? 0 7(1) - 5 - 2 ? 0 0 = 0 So we verified our answer, [B]x = -1[/B].

if a number is added to its square, it equals 20. Let the number be an arbitrary variable, let's call it n. The square of the number means we raise n to the power of 2: n^2 We add n^2 to n: n^2 + n It equals 20 so we set n^2 + n equal to 20 n^2 + n = 20 This is a quadratic equation. So [URL='https://www.mathcelebrity.com/quadratic.php?num=n%5E2%2Bn%3D20&pl=Solve+Quadratic+Equation&hintnum=+0']we type this equation into our search engine[/URL] to solve for n and we get two solutions: [B]n = (-5, 4)[/B]

If x2 is added to x, the sum is 42. x^2 + x = 42 Subtract 42 from both sides: x^2 + x - 42 = 0 We have a quadratic equation. Using our [URL='http://www.mathcelebrity.com/quadratic.php?num=x%5E2%2Bx-42%3D0&pl=Solve+Quadratic+Equation&hintnum=+0']quadratic equation solver[/URL], we get: [B]x = 6 and x = -7 [/B] Since the problem does not state positive number solutions, they are both answers.

Is (1, 3) a solution to the equation y = 3x? Plug in x = 1 into y = 3x: y = 3(1) y = 3 The answer is [B]yes[/B], (1, 3) is a solution to y = 3x

Is (3,10) a solution to the equation y=4x Plug in the numbers to check: 10 ? 4(3) 10 <> 12 No, this is [B]not a solution[/B]

Is (9, 6) a solution to the equation y = x - 3? The ordered pair (x, y) = (9, 6) Plug in x = 9 into y = x - 3: y = 9 - 3 y = 6 [B]Yes, (9, 6) is a solution to the equation y = x - 3[/B]

Is 30 a solution to 2x + 5 = 3x - 25 Let's test x = 30 into our equation: 2(30) + 5 ? 3(30) - 25 60 + 5 ? 90 - 25 65 = 65 [B]Yes, x = 30 is a solution[/B]. If you wanted to solve for x with simplification, you can [URL='https://www.mathcelebrity.com/1unk.php?num=2x%2B5%3D3x-25&pl=Solve']type it in our search engine[/URL] and get: x = 30

Is the point (4,7) a solution of the equation y equals 15x minus 8? Plug in x = 4: 15(4) - 8 60 - 8 52 Since 52 <> 4, (4,7) is [U][B]not[/B][/U] a solution of the equation y equals 15x minus 8

Kamille is calculating the length of diagonal on a picture board and gets a solution of the square root of 58. She needs to buy the ribbon to put across the diagonal of the board, so she estimates that she will need at least 60 inches of ribbon to cover the diagonal. Is she correct? Explain. [URL='https://www.mathcelebrity.com/powersq.php?num=sqrt%2858%29&pl=Calculate']The square root of 58 [/URL]has an answer between 7 and 8. So Kamille is [B]incorrect[/B]. She needs much less than 60 inches of ribbon. She needs less than 8 inches of ribbon.

Kevin ran 4 miles more than Steve ran. The sum of their distances is 26 miles. How far did Steve run? The domain of the solution is: Let k be Kevin's miles ran Let s be Steve's miles ran We have 2 given equtaions: [LIST=1] [*]k = s + 4 [*]k + s = 26 [/LIST] Substitute (1) into (2) (s + 4) + s = 26 2s + 4 = 26 Plug this into our [URL='http://www.mathcelebrity.com/1unk.php?num=2s%2B4%3D26&pl=Solve']equation calculator[/URL] and we get s = 11

Free Linear Congruence Calculator - Given an modular equation ax ≡ b (mod m), this solves for x if a solution exists

Contains word problems and other long form problems with step-by-step solutions.

Free Molar Mass Calculator - Calculates the molar mass of an element or solution.

One number is equal to the square of another. Find the numbers if both are positive and their sum is 650 Let the number be n. Then the square is n^2. We're given: n^2 + n = 650 Subtract 650 from each side: n^2 + n - 650 = 0 We have a quadratic equation. [URL='https://www.mathcelebrity.com/quadratic.php?num=n%5E2%2Bn-650%3D0&pl=Solve+Quadratic+Equation&hintnum=+0']We type this into our search engine[/URL] and we get: n = 25 and n = -26 Since the equation asks for a positive solution, we use [B]n = 25[/B] as our first solution. the second solution is 25^2 = [B]625[/B]

Penny bought a new car for $25,000. The value of the car has decreased in value at rate of 3% each year since. Let x = the number of years since 2010 and y = the value of the car. What will the value of the car be in 2020? Write the equation, using the variables above, that represents this situation and solve the problem, showing the calculation you did to get your solution. Round your answer to the nearest whole number. We have the equation y(x): y(x) = 25,000(0.97)^x <-- Since a 3 % decrease is the same as multiplying the starting value by 0.97 The problem asks for y(2020). So x = 2020 - 2010 = 10. y(10) = 25,000(0.97)^10 y(10) = 25,000(0.73742412689) y(10) = [B]18,435.60[/B]

Quadratic equation hacks using the discriminant Solve x^2- 4x+ 5 using a discriminant: Discriminant is: Discriminant = b^2- 4ac Discriminant = (-4)^2 - 4(1)(5) Discriminant = 16 - 20 Discriminant = -4 When Discriminant < 0, the quadratic has [I][U]no solution [MEDIA=youtube]RogZ3430_8E[/MEDIA][/U][/I]

Free Solution Mixture Calculator - Determines a necessary amount of a Solution given two solution percentages and 1 solution amount.

The difference between two positive numbers is 5 and the square of their sum is 169. Let the two positive numbers be a and b. We have the following equations: [LIST=1] [*]a - b = 5 [*](a + b)^2 = 169 [*]Rearrange (1) by adding b to each side. We have a = b + 5 [/LIST] Now substitute (3) into (2): (b + 5 + b)^2 = 169 (2b + 5)^2 = 169 [URL='https://www.mathcelebrity.com/community/forums/calculator-requests.7/create-thread']Run (2b + 5)^2 through our search engine[/URL], and you get: 4b^2 + 20b + 25 Set this equal to 169 above: 4b^2 + 20b + 25 = 169 [URL='https://www.mathcelebrity.com/quadratic.php?num=4b%5E2%2B20b%2B25%3D169&pl=Solve+Quadratic+Equation&hintnum=+0']Run that quadratic equation in our search engine[/URL], and you get: b = (-9, 4) But the problem asks for [I]positive[/I] numbers. So [B]b = 4[/B] is one of our solutions. Substitute b = 4 into equation (1) above, and we get: a - [I]b[/I] = 5 [URL='https://www.mathcelebrity.com/1unk.php?num=a-4%3D5&pl=Solve']a - 4 = 5[/URL] [B]a = 9 [/B] Therefore, we have [B](a, b) = (9, 4)[/B]

The length of a wooden frame is 1 foot longer than its width and its area is equal to 12ft The frame is a rectangle. The area of a rectangle is A = lw. So were given: [LIST=1] [*]l = w + 1 [*]lw = 12 [/LIST] Substitute equation (1) into equation (2) for l: (w + 1) * w = 12 Multiply through and simplify: w^2 + w = 12 We have a quadratic equation. To solve for w, we type this equation into our search engine and we get two solutions: w = 3 w = -4 Since width cannot be negative, we choose the positive result and have: w = [B]3[/B] To solve for length, we plug w = 3 into equation (1) above and get: l = 3 + 1 l = [B]4[/B]

The point (1,5) is a solution to the equation 2y - x = 9 [B]Yes[/B], because: 2(5) - 1 ? 9 10 - 1 ? 9 9 = 9

Write this out, let the number be x. sqrt(2x) = x - 4 since 4 less means subtract Square each side: sqrt(2x)^2 = (x - 4)^2 2x = x^2 - 8x + 16 Subtract 2x from both sides x^2 - 10x + 16 = 0 Using the [URL='http://www.mathcelebrity.com/quadratic.php?num=x%5E2+-+10x+%2B+16+%3D+0&pl=Solve+Quadratic+Equation&hintnum=0']quadratic calculator[/URL], we get two potential solutions x = (2, 8) Well, 2 does not work, since sqrt(2*2) = 2 which is not 4 less than 2 However, 8 does work: sqrt(2*8) = sqrt(16) = 4, which is 4 less than the number 8.

The sum of a number and its square is 72. find the numbers? Let the number be n. We have: n^2 + n = 72 Subtract 72 from each side: n^2 + n - 72 = 0 Using our [URL='http://www.mathcelebrity.com/quadratic.php?num=n%5E2%2Bn-72%3D0&pl=Solve+Quadratic+Equation&hintnum=+0']quadratic calculator[/URL], we have: [B]n = 8 or n = -9 [/B] Since the numbers do not state positive or negative, these are the two solutions.

The temperature of a solution was -23C. After adding a substance to the solution, the temperature after adding the substance to the solution was 133C. What is the difference between the temperature of the solution before and after adding the substance Using our [URL='https://www.mathcelebrity.com/temp-change.php?num=thetemperatureofasolutionwas-23c.afteraddingasubstancetothesolutionthetemperaturefe133c.whatisthedifferencebetweenthetemperatureofthesolutionbeforeandafteraddingthesubstance%3E&pl=Calculate+Temp+Change']temperature difference calculator[/URL], we get: [B]156C[/B]

Translate and solve: 30 times m is greater than ?330. (Write your solution in interval notation.) 30 times m: 30m is greater than -330 30m > -330 Using our [URL='https://www.mathcelebrity.com/interval-notation-calculator.php?num=30m%3E-330&pl=Show+Interval+Notation']equation and interval solver[/URL], we get: m > -11

When 28 is subtracted from the square of a number, the result is 3 times the number. Find the negative solution. Let the number be n. Square of a number: n^2 28 is subtracted from the square of a number: n^2 - 28 3 times the number: 3n [I]The result is[/I] mean an equation, so we set n^2 - 28 = 3n n^2 - 28 = 3n Subtract 3n from each side: n^2 - 3n - 28 = 3n - 3n The right side cancels to 0, so we have: n^2 - 3n - 28 = 0 This is a quadratic equation in standard form, so we [URL='https://www.mathcelebrity.com/quadratic.php?num=n%5E2-3n-28%3D0&pl=Solve+Quadratic+Equation&hintnum=+0']use our quadratic calculator[/URL] to solve: We get two solutions for n: n = (-4, 7) The question asks for the negative solution, so our answer is: [B]n = -4[/B]

When 4 is subtracted from the square of a number, the result is 3 times the number. Find the positive solution. Let the number be n. We have: n^2 - 4 = 3n Subtract 3n from each side: n^2 - 3n - 4 = 0 [URL='https://www.mathcelebrity.com/quadratic.php?num=n%5E2-3n-4%3D0&pl=Solve+Quadratic+Equation&hintnum=+0']Typing this quadratic equation into the search engine[/URL], we get: n = (-1, 4) The problem asks for the positive solution, so we get [B]n = 4[/B].

Which of the following is the probability that subjects do not have the disease, but the test result is positive? a. Miss rate b. False positive rate c. Base rate d. Disease rate [B]b. [URL='http://sites.stat.psu.edu/~lsimon/stat250/sp00/solutions/misc/diagtests.htm']False positive rate[/URL][/B]

ncG1vNJzZmivp6x7rq3ToZqepJWXv6rA2GeaqKVfqLKivsKhZamgoHS%2Bfr%2FOpaytoZ%2Bj