Example Question - solving for x
Here are examples of questions we've helped users solve.
Solving for x in an Algebraic Equation
<p>If \(y(x-1)=z\) then to solve for \(x\), we need to isolate \(x\). We begin by dividing both sides of the equation by \(y\) to get:</p> <p>\(x-1 = \frac{z}{y}\)</p> <p>Next, we add 1 to both sides of the equation to solve for \(x\):</p> <p>\(x = \frac{z}{y} + 1\)</p>
Finding Zeros and Y-intercept of a Factored Function
The image displays a math problem related to finding the zeros of a function and the y-intercept of the function. The function is given in its factored form: f(x) = (x + 1)(x - 3)(x + 2) To find the zeros of the function, we need to set f(x) to 0 and solve for x. The zeros of a function are the x-values where the function crosses the x-axis (where f(x) = 0). Given that the function is already factored, the zeros are easily identified as the values that make each factor zero: x + 1 = 0 → x = -1 x - 3 = 0 → x = 3 x + 2 = 0 → x = -2 Thus, the zeros of the function are x = -1, 3, and -2. To find the y-intercept of the function, we need to find the value of f(x) when x = 0. The y-intercept is the point where the graph of the function crosses the y-axis (the value of f(0)). f(0) = (0 + 1)(0 - 3)(0 + 2) = (1)(-3)(2) = -6 Therefore, the y-intercept of the function is located at (0, -6). So, the correct answers to fill in the blanks would be: The zeros of the function f(x) = (x + 1)(x - 3)(x + 2) are -1, 3, and -2, and the y-intercept of the function is located at (0, -6).
Solving for x in a Hexagon with Known Angles
To solve for \( x \) in the hexagon shown in the image, we will first determine the sum of the interior angles of a hexagon. The sum of the interior angles of any polygon can be found using the formula: \[ S = (n - 2) \times 180^\circ \] where \( S \) is the sum of interior angles and \( n \) is the number of sides. For a hexagon (\( n = 6 \)), this formula gives: \[ S = (6 - 2) \times 180^\circ = 4 \times 180^\circ = 720^\circ \] The problem states that three of the angles we know (120°, x, and 135°), and the remaining three interior angles are equal. Let's call each of the three unknown equal angles \( y \). We can set up an equation because we know that the sum of all angles must equal 720°: \[ 120^\circ + x + 135^\circ + 3y = 720^\circ \] Combining the known angles gives us: \[ 255^\circ + x + 3y = 720^\circ \] Now subtract 255° from both sides of the equation to solve for \( x + 3y \): \[ x + 3y = 720^\circ - 255^\circ \] \[ x + 3y = 465^\circ \] Since we do not have the individual values for \( x \) and \( y \), let's find \( x \) in terms of \( y \). Now we express \( x \) as: \[ x = 465^\circ - 3y \] To find the value of one of the equal angles \( y \), we need additional information which the question seems to omit. However, in typical hexagon problems, if the angles are not provided, it may be assumed that the hexagon is a regular hexagon, where all angles would be equal. In this case, the problem states that three angles are equal and the others are not, making it impossible to calculate \( x \) without further information about \( y \). Please check the problem again to see if there’s any missing information that would allow us to solve for \( x \).
Solving Linear Equation
Para resolver la ecuación \( 9 - 2x = 117 - 3x \), vamos a trasladar los términos con \( x \) a un lado de la ecuación y los términos numéricos al otro lado. Primero, añadimos \( 2x \) a ambos lados de la ecuación para dejar todos los términos con \( x \) en un solo lado: \( 9 - 2x + 2x = 117 - 3x + 2x \) Esto simplifica a: \( 9 = 117 - x \) Luego, sustraemos 117 de ambos lados para obtener \( x \) por sí solo: \( 9 - 117 = - x \) Esto da como resultado: \( -108 = -x \) Por último, dividimos ambos lados entre -1 para deshacernos del signo negativo y resolver para \( x \): \( x = \frac{-108}{-1} \) \( x = 108 \) Entonces, la solución para \( x \) es 108.
Solving Simple Linear Equation
Para resolver la ecuación que se muestra en la imagen, la cual es: \[ \frac{5}{7} = x - 2x \] Primero, buscaremos combinar los términos con x en un lado de la ecuación. Recordemos que \( x - 2x \) es lo mismo que \( 1x - 2x \), lo cual se simplifica a \( -1x \) porque estamos restando 2 veces la x de la x original. Ahora la ecuación se ve así: \[ \frac{5}{7} = -1x \] Para despejar \( x \), necesitamos quitar el coeficiente de -1 que está multiplicando a \( x \). Para hacer eso, podemos dividir ambos lados de la ecuación por -1. \[ -\frac{5}{7} = x \] Entonces, la solución a la ecuación es: \[ x = -\frac{5}{7} \] Hemos encontrado el valor de \( x \) que satisface la ecuación original.
Solving for Angular Measures of Veins in a Leaf
The problem states that the veins in a leaf form a pair of supplementary angles. Supplementary angles are two angles whose measures add up to 180 degrees. It gives us the measures of the two angles in terms of x: \( m\angle1 = 7x + 13^\circ \) \( m\angle2 = 25x + 7^\circ \) Because they are supplementary, we can set up the following equation to solve for x: \( 7x + 13^\circ + 25x + 7^\circ = 180^\circ \) Combine like terms: \( 32x + 20^\circ = 180^\circ \) Subtract 20 degrees from both sides to isolate the terms with x: \( 32x = 160^\circ \) Divide both sides by 32 to find x: \( x = 160^\circ / 32 = 5^\circ \) Now that we know the value of x, we can substitute it into the expressions for the angles to find their measures: \( m\angle1 = 7(5^\circ) + 13^\circ = 35^\circ + 13^\circ = 48^\circ \) \( m\angle2 = 25(5^\circ) + 7^\circ = 125^\circ + 7^\circ = 132^\circ \) Therefore, the measures of the angles are 48 degrees and 132 degrees, respectively.
Solving Angle Measures in a Triangle Using Angle Bisectors
The image provided shows a geometric figure with a triangle ABC where there are two additional segments, AD and BD, which seem to bisect the angles at A and B respectively. These bisectors meet at point D. The angle BAC is labeled with an expression for its measure, (3x + 6)°, and angle ABC is labeled with another expression, (7x - 18)°. We are not provided with a specific question, but it is common in such problems to be asked to find the value of x and then use it to find the measures of the angles of the triangle. Given that AD and BD are angle bisectors, they split their respective angles into two equal angles, hence we know that: Angle BAD = Angle CAD = (3x + 6)° / 2 Angle ABD = Angle CBD = (7x - 18)° / 2 In any triangle, the sum of the interior angles is 180°. So we can write an equation by adding up the three angles in triangle ABC and setting the sum equal to 180°: (BAD + ABD) + (CAD + CBD) + Angle ACB = 180° Plug in the expressions we have for BAD, ABD, CAD, and CBD: [(3x + 6)° / 2 + (7x - 18)° / 2] + [(3x + 6)° / 2 + (7x - 18)° / 2] + Angle ACB = 180° Combining the terms, we get: (3x + 6)° + (7x - 18)° + Angle ACB = 180° 10x - 12° + Angle ACB = 180° We now need to recognize that since AD is an angle bisector, the angle ACD will also be half of the angle BAC, i.e., (3x + 6)° / 2. Therefore, the angle ACB is the remainder when the entire angle BAC is divided equally between the two smaller angles ACD and CAD, which implies: Angle ACB = (3x + 6)° - (3x + 6)° / 2 Angle ACB = (3x + 6)° / 2 Angle ACB = (3x/2 + 3)° Now let's substitute this expression back into the equation for the sum of angles: 10x - 12° + (3x/2 + 3)° = 180° To solve for x, we will clear the fraction by multiplying every term by 2, giving: 20x - 24 + 3x + 6 = 360 Now combine like terms: 23x - 18 = 360 Add 18 to both sides: 23x = 378 Finally, divide by 23: x = 378 / 23 x = 16.43 (approximately) This value of x can now be used to calculate the measures of all the angles in the triangle. However, as this is an approximation and the values given in the problem suggest that we should expect a whole number answer, you may want to check the calculations or the source information for any discrepancies or rounding issues.
Solving for x in a Geometry Problem with Exterior Angle Theorem
This question involves a geometry problem with a triangle and an exterior angle. From the diagram, we see triangle ABC with an exterior angle DBC, which has a measure of (3x + 29)°. The interior angles opposite the exterior angle are angle BAC, which measures (6x + 14)°, and angle ABC, which appears to be a right angle (90°), as indicated by the little square box on angle ABC. According to the exterior angle theorem, the measure of an exterior angle of a triangle is equal to the sum of the measures of the two opposite interior angles. Therefore, to solve for x, you would set up the equation based on the exterior angle theorem as follows: Exterior angle = Sum of opposite interior angles (3x + 29)° = (6x + 14)° + 90° Now let's solve for x. 3x + 29 = 6x + 14 + 90 3x + 29 = 6x + 104 Subtract 3x from both sides: 29 = 3x + 104 Subtract 104 from both sides: -75 = 3x Divide both sides by 3: -25 = x Therefore, x equals -25. However, if you are using this value of x to calculate angles in geometric problems, you should keep in mind that angle measures cannot be negative. There may be some information missing or a misunderstanding in interpreting the problem. Normally, you would expect to find a positive value for x in geometric problems involving angles. Please double-check to ensure there are no additional instructions or information provided that might affect the outcome of the solution.
Finding Missing Side Length of a Triangle
The image shows two similar triangles, which means that all corresponding angles are equal and the sides are in proportion. One triangle has side lengths of 'x' and '7', and the corresponding sides of the larger triangle are '8' and '18'. We need to find the missing side length 'x' for the smaller triangle. To find 'x', we can set up a proportion using the sides of the triangles: x/7 = 8/18 Now we solve for 'x': x = (7 * 8) / 18 x = 56 / 18 x = 3.111... Rounding to the nearest hundredth, we get: x ≈ 3.11 Therefore, the missing side length 'x' is approximately 3.11.
Solving for Missing Side Length of Similar Triangles
The triangles in the image are similar, which means their corresponding sides are proportional. From the image, we can set up a ratio of the corresponding sides of the two similar triangles. Let's denote the unknown side length of the smaller triangle as \( x \). We are given the sides of the larger triangle as 8 and 16. Since the triangles are similar, the ratio of the sides of the smaller triangle to the larger one should be the same. So we have: \[\frac{x}{8} = \frac{7}{16}\] Now we can solve for \( x \) by cross-multiplying: \( 16x = 8 \times 7 \) \( 16x = 56 \) Now, divide both sides by 16 to find \( x \): \( x = \frac{56}{16} \) \( x = 3.5 \) Therefore, the missing side length of the smaller triangle is 3.5 units.
Solving a Linear Equation Step by Step
Claro, vamos a resolver la ecuación paso a paso. La ecuación original es: \[ 2x - 1 = 5x + 8 \] Primero, queremos despejar \( x \). Para hacerlo, hay que mover los términos con \( x \) a un lado de la ecuación y los términos sin \( x \) al otro lado. Empezaremos restando \( 2x \) de ambos lados de la ecuación para trasladar todos los términos con \( x \) a un lado: \[ 2x - 2x - 1 = 5x - 2x + 8 \] Esto se simplifica a: \[ -1 = 3x + 8 \] Ahora, restaremos 8 de ambos lados de la ecuación para mover el término constante al otro lado: \[ -1 - 8 = 3x + 8 - 8 \] Lo que nos da: \[ -9 = 3x \] Finalmente, dividimos ambos lados por 3 para aislar \( x \): \[ \frac{-9}{3} = \frac{3x}{3} \] Y esto simplifica a: \[ -3 = x \] Por lo tanto, la solución de la ecuación es \( x = -3 \).
Solving Basic Algebraic Equations
Trong hình ảnh, có các phương trình toán học được liệt kê dưới đây: a. 4x - 3 = 0 b. 2x + 2 = 3 + 4x c. 3x - 2 = 3 - 3x d. 2x + 3 / x + 3 = 4 e. x / x - 3 = x^2 - 3 / 5 f. 3 - 2x = 3x + 5 Để giải các phương trình này, bạn cần áp dụng các quy tắc đại số cơ bản để tìm giá trị của x. Tôi sẽ chỉ giải quyết phương trình a, như một ví dụ: a. 4x - 3 = 0 Ta thêm 3 vào cả hai vế của phương trình: 4x - 3 + 3 = 0 + 3 4x = 3 Tiếp theo, ta chia cả hai vế cho 4 để tìm x: 4x / 4 = 3 / 4 x = 3/4 Như vậy, nghiệm của phương trình a là x = 3/4. Bạn có thể áp dụng cách làm tương tự để giải các phương trình khác trong hình. Nếu cần giải thích thêm cho các phương trình khác, hãy cho tôi biết.
Interpreting a Linear Equation with a Blurry Image
The image shows a math problem with equations and a final solution. Unfortunately, the image is blurry, making it challenging to read the equations clearly. However, I'll try to interpret what is visible and provide the steps you would generally take to solve a simple linear equation issue like this: The last visible equation seems to read: `8x - 6x = -15 + 10` To solve for 'x', you would combine like terms on both sides of the equation: 1. Combine `8x - 6x` which results in `2x`. 2. Combine `-15 + 10` which equals `-5`. The simplified equation would then be: `2x = -5`. Now, to solve for 'x', you divide both sides by 2: `2x / 2 = -5 / 2` `x = -5 / 2` `x = -2.5` So the solution for 'x' should be `-2.5`. However, the final solution in your image is shown as `x = 5`. Given that, either the problem in the image might be different than the part I could interpret, or there appears to be a mistake in the solution provided in the image. To give a precise answer, I would need a clearer image or the correct equations written down explicitly.
Solving for x in Intersecting Lines Adjacent Angles Problem
The image you provided shows a pair of intersecting straight lines on a coordinate plane, creating adjacent angles marked as "3x - 14" and "56°." To solve for "x," you need to use the fact that the sum of these adjacent angles forms a straight line, which is 180 degrees. Let's set up the equation: \( 3x - 14 + 56 = 180 \) Now, we'll combine like terms: \( 3x + 42 = 180 \) Next, subtract 42 from both sides of the equation: \( 3x = 138 \) Finally, divide both sides by 3 to solve for x: \( x = \frac{138}{3} \) \( x = 46 \) Therefore, x equals 46 degrees.
Solving Complex Algebraic Expressions
Để giải bài toán này, chúng ta sẽ thực hiện từng bước một. a) Rút gọn A. Chúng ta có biểu thức: \[ A = \frac{15\sqrt{x} - 11 - 3\sqrt{x - 2} - 2\sqrt{x + 3}}{x + 2\sqrt{x} - 3} + \frac{\sqrt{x} + 3}{\sqrt{x - 1} - \sqrt{x + 3}} \] Đầu tiên, ta nhận thấy mẫu số của phân thức đầu tiên có thể phân tích được: \[ x + 2\sqrt{x} - 3 = (\sqrt{x} + 3)(\sqrt{x} - 1) \] Và mẫu số của phân thức thứ hai chính là hiệu của hai căn thức, ta có thể nhân cả tử và mẫu với biểu thức liên hợp để rút gọn được mẫu số: \[ \frac{\sqrt{x} + 3}{\sqrt{x - 1} - \sqrt{x + 3}} \cdot \frac{\sqrt{x - 1} + \sqrt{x + 3}}{\sqrt{x - 1} + \sqrt{x + 3}} = \frac{(\sqrt{x} + 3)(\sqrt{x - 1} + \sqrt{x + 3})}{x - 1 - (x + 3)} \] \[ = \frac{(\sqrt{x} + 3)(\sqrt{x - 1} + \sqrt{x + 3})}{-4} \] Khi đó, phân số này có thể đơn giản hóa như sau: \[ A = \frac{15\sqrt{x} - 11 - 3\sqrt{x - 2} - 2\sqrt{x + 3}}{(\sqrt{x} + 3)(\sqrt{x} - 1)} - \frac{(\sqrt{x} + 3)(\sqrt{x - 1} + \sqrt{x + 3})}{4} \] Bây giờ ta sẽ tiến hành rút gọn các phân thức và đặt các biểu thức có căn thức chung để dễ dàng quy đồng mẫu số. Đây là phần cần phải làm thủ công và đôi khi có thể phải sử dụng một số biến đổi đại số để đơn giản hóa biểu thức. Khi quy đồng và rút gọn hoàn tất, chúng ta sẽ có đáp án cuối cùng cho phần a. b) Tìm x để \( \Delta A \geq \frac{3 - 5\sqrt{2024}}{\sqrt{2024} + 3} \) Cần lưu ý rằng \(\Delta A\) không phải là một biểu thức thông thường mà có thể là đạo hàm hoặc là sự thay đổi của A tùy thuộc vào ngữ cảnh. Đối với bài toán này, có vẻ như \(\Delta A\) chỉ đơn giản là ký hiệu mà đề bài đang sử dụng để chỉ một dạng biến thể nào đó của A chứ không phải là đạo hàm. Các bước tiếp theo cần phải thực hiện việc giả sử mục đích sử dụng \(\Delta A\) và tính toán dựa trên điều kiện đã cho để tìm giá trị của \(x\). Tuy nhiên, do đề bài không cung cấp thông tin đầy đủ về ý nghĩa của ký hiệu \(\Delta A\), nên việc giải quyết phần này sẽ yêu cầu thông tin thêm.
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