Example Question - linear equation
Here are examples of questions we've helped users solve.
Algebraic Fraction Equation Problem
<p>Умножьте обе стороны уравнения на 12 (наименьшее общее кратное знаменателей), чтобы избавиться от знаменателей:</p> <p>\( 12 \times \frac{x-3y}{4} = 12 \times \frac{3y-2x}{6} \)</p> <p>\( 3(x-3y) = 2(3y-2x) \)</p> <p>\( 3x - 9y = 6y - 4x \)</p> <p>Перенесите все члены уравнения в одну сторону:</p> <p>\( 3x + 4x = 9y + 6y \)</p> <p>\( 7x = 15y \)</p> <p>Теперь, разделите обе стороны на 7, чтобы найти \( y \) в терминах \( x \):</p> <p>\( x = \frac{15}{7}y \)</p> <p>\( y = \frac{7}{15}x \)</p>
Solving a Linear Equation with Fractions
<p>La ecuación original es \( y = 7 - 2 \cdot (\frac{2}{3}) \).</p> <p>Primero, simplificamos la operación dentro del paréntesis:</p> <p>\( y = 7 - 2 \cdot \frac{2}{3} \)</p> <p>\( y = 7 - \frac{4}{3} \)</p> <p>Luego, convertimos el número entero 7 a una fracción con denominador 3 para poder hacer la resta:</p> <p>\( y = \frac{21}{3} - \frac{4}{3} \)</p> <p>Finalmente, realizamos la resta de fracciones con el mismo denominador:</p> <p>\( y = \frac{21 - 4}{3} \)</p> <p>\( y = \frac{17}{3} \)</p>
Graphing a Linear Equation
<p>Para graficar la ecuación lineal \(2x + 3y = 7\), primero encontramos dos puntos resolviendo para una variable mientras asignamos valores a la otra.</p> <p>Si \(x=0\): \[2(0) + 3y = 7 \] \[3y = 7 \] \[y = \frac{7}{3} \] Entonces, tenemos el punto \((0, \frac{7}{3})\).</p> <p>Si \(y=0\): \[2x + 3(0) = 7 \] \[2x = 7 \] \[x = \frac{7}{2} \] Entonces, tenemos el punto \((\frac{7}{2}, 0)\).</p> <p>Con estos dos puntos, podemos trazar la recta que representa la ecuación dada.</p>
Graphing a Linear Equation
<p>Para graficar la ecuación lineal 2x + 3y = 7, primero encontramos los interceptos en los ejes x e y.</p> <p>Intercepto en x cuando y = 0:</p> <p>2x + 3(0) = 7 \Rightarrow 2x = 7 \Rightarrow x = \frac{7}{2}</p> <p>Intercepto en y cuando x = 0:</p> <p>2(0) + 3y = 7 \Rightarrow 3y = 7 \Rightarrow y = \frac{7}{3}</p> <p>Con los puntos (\frac{7}{2},0) y (0,\frac{7}{3}) podemos dibujar la recta.</p>
Graphing a Linear Equation on a Coordinate Plane
<p>Para resolver esta pregunta, necesitamos graficar la ecuación lineal \(y = 3x - 2\) en el plano de coordenadas. Primero, utilizando la tabla de valores, podemos ver algunos puntos que ya se han calculado y que se pueden trazar en la gráfica.</p> <p>Paso 1: Trazar los puntos (3, 7), (2, 4) y (-2, -8) en el plano de coordenadas. Cada punto corresponde a un valor de 'x' y el valor de 'y' resultante tras aplicar la ecuación \(y = 3x - 2\).</p> <p>Paso 2: Dibujar una línea recta que pase por estos puntos, ya que representan la solución a la ecuación lineal y cualquier punto en esta línea satisfará la ecuación \(y = 3x - 2\).</p> <p>El punto donde la línea cruza el eje 'y' es el intercepto en y, que para esta ecuación es -2, y esto indica que cuando \(x=0\), \(y=-2\).</p> <p>La pendiente de la línea es 3, indicando que por cada aumento en 1 en la dirección de 'x', 'y' aumentará en 3 unidades.</p>
Graphing a Linear Equation
\[ \begin{array}{c} \text{Para el valor de} \ x = 3: \\ y = 3(3) - 2 = 9 - 2 = 7 \\ \text{Por lo tanto, el par ordenado es} \ (3,7). \\ \text{Para el valor de} \ x = 2: \\ y = 3(2) - 2 = 6 - 2 = 4 \\ \text{Por lo tanto, el par ordenado es} \ (2,4). \\ \text{Para el valor de} \ x = -2: \\ y = 3(-2) - 2 = -6 - 2 = -8 \\ \text{Por lo tanto, el par ordenado es} \ (-2,-8). \\ \text{Para el valor de} \ x = 0: \\ y = 3(0) - 2 = 0 - 2 = -2 \\ \text{Por lo tanto, el par ordenado es} \ (0,-2). \\ \end{array} \] Con estos pares ordenados, se pueden marcar los puntos correspondientes en el plano coordenado y trazar la recta que los une para representar la ecuación lineal \( y = 3x - 2 \).
Solving a Linear Equation
<p>\(2x^2 - 10356 = 99800\)</p> <p>Đầu tiên, chúng ta cần chuyển phương trình về dạng chuẩn của phương trình bậc hai:</p> <p>Đưa cả hai vế về cùng một phía:</p> <p>\(2x^2 - 10356 - 99800 = 0\)</p> <p>Giải phương trình bậc hai:</p> <p>\(2x^2 - 110156 = 0\)</p> <p>Chia cả hai vế cho 2:</p> <p>\(x^2 - 55078 = 0\)</p> <p>Thêm 55078 vào cả hai vế:</p> <p>\(x^2 = 55078\)</p> <p>Lấy căn bậc hai cho cả hai vế:</p> <p>\(x = \pm \sqrt{55078}\)</p> <p>Vậy phương trình có hai nghiệm là:</p> <p>\(x = \sqrt{55078}\) hoặc \(x = -\sqrt{55078}\)</p>
Solving a Linear Equation for a Given Variable
<p>La question montre une équation linéaire de la forme \( P(x) = 0.006x + 350 \).</p> <p>Pour résoudre cette équation pour une valeur spécifique de \( P(x) \), on doit remplacer \( P(x) \) par cette valeur et résoudre l'équation pour \( x \). Malheureusement, la valeur spécifique de \( P(x) \) n'est pas donnée dans l'image. Si une valeur est donnée, disons \( P(x) = k \), alors l'équation serait résolue comme suit:</p> <p>\( k = 0.006x + 350 \)</p> <p>\( x = \frac{k - 350}{0.006} \)</p> <p>Cela donne la valeur de \( x \) en fonction de \( k \). Sans une valeur spécifique pour \( P(x) \), nous ne pouvons pas procéder à une résolution numérique précise.</p>
Solving a Linear Equation with Distributed Terms
<p>The given equation is:</p> <p>\(\frac{1}{2} (90 + 22x - 10) \cdot 120 = 14,400\).</p> <p>To solve for \(x\), first simplify the equation:</p> <p>Multiply 120 by each term inside the parentheses:</p> <p>\(\frac{1}{2} (120 \cdot 80 + 120 \cdot 22x) = 14,400\)</p> <p>\(\frac{1}{2} (9600 + 2640x) = 14,400\)</p> <p>Multiply both sides by 2 to eliminate the fraction:</p> <p>\(9600 + 2640x = 28,800\)</p> <p>Subtract 9600 from both sides to isolate the \(x\) term:</p> <p>\(2640x = 28,800 - 9600\)</p> <p>\(2640x = 19,200\)</p> <p>Divide both sides by 2640 to solve for \(x\):</p> <p>\(x = \frac{19,200}{2640}\)</p> <p>\(x = 7.2727...\)</p> <p>As an exact fraction, \(x\) equals:</p> <p>\(x = \frac{7}{22}\)</p>
Solve the Linear Equation for the Unknown Variable
<p>Given equation: \( \frac{1}{2} (70 + 2x -10)(20) = \frac{1}{3}600 \)</p> <p>Multiply through by 2 to get rid of the fraction: \( (70 + 2x - 10)(20) = \frac{2}{3}600 \)</p> <p>Simplify inside the parentheses: \( (60 + 2x)(20) = \frac{2}{3}600 \)</p> <p>Multiply out the parentheses: \( 1200 + 40x = \frac{2}{3}600 \)</p> <p>Divide both sides by 40: \( x + 30 = \frac{2}{3} \times 15 \)</p> <p>Divide 600 by 3: \( x + 30 = \frac{2}{3} \times 15 = 2 \times 5 = 10 \)</p> <p>Subtract 30 from both sides: \( x = 10 - 30 \)</p> <p>Therefore: \( x = -20 \)</p>
Analysis of Coordinates and Linear Equation of Points in Cartesian Plane
<p>Given point Q(8, k) and point S(-6, 0), it is known that line PQ is parallel to the x-axis.</p> <p>(a) Since PQ is parallel to the x-axis, the y-coordinates of points P and Q are equal, which means the y-coordinate of point Q is k = 0 (the same as point S).</p> <p>\(k = 0\)</p> <p>(b) To find the equation of line PS, we can calculate the slope (m) using the coordinates of points P and S:</p> <p>\(m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{0 - 0}{-6 - 8}\)</p> <p>Since the change in y is 0 (the line is horizontal), the slope is:</p> <p>\(m = 0\)</p> <p>The general equation of a line is \(y = mx + b\). Since the slope m = 0, the equation simplifies to</p> <p>\(y = b\)</p> <p>To find b, we use the fact that the line passes through point S(-6, 0):</p> <p>\(0 = 0 \cdot (-6) + b\)</p> <p>\(b = 0\)</p> <p>The equation of line PS is therefore \(y = 0\).</p>
Solving a Linear Equation with Multiple Terms
<p>First, expand the terms: \( 2(3x + 5) - 3(3x - 1) = 3(4 - x) \)</p> <p>\( 6x + 10 - 9x + 3 = 12 - 3x \)</p> <p>Simplify the equation: \( -3x + 13 = 12 - 3x \)</p> <p>Since the \(x\)-terms are on both sides and they cancel each other out, we're left with:</p> <p>\( 13 = 12 \)</p> <p>This is clearly not true; therefore, there is no solution for this equation, and the lines represented by these equations would be parallel and never intersect.</p>
Solving a Linear Equation with Variables on Both Sides
<p>Begin by expanding the terms within the parentheses:</p> <p>\[2(3x + 5) - 3(3x - 1) = 3(4 - x)\]</p> <p>\[6x + 10 - 9x + 3 = 12 - 3x\]</p> <p>Combine like terms:</p> <p>\[-3x + 13 = 12 - 3x\]</p> <p>Add \(3x\) to both sides to move variables to one side:</p> <p>\[-3x + 3x + 13 = 12 - 3x + 3x\]</p> <p>\[13 = 12\]</p> <p>Since 13 is not equal to 12, we have an inconsistency. The equation has no solution because it is not an identity, and the two sides of the equation are not equal for any value of \(x\).</p>
Linear Equation Single Variable Problem
<p>\( 6 - 5x = 13 \)</p> <p>\( -5x = 13 - 6 \)</p> <p>\( -5x = 7 \)</p> <p>\( x = -\frac{7}{5} \)</p> <p>\( x = -1.4 \)</p>
Solving the Linear Equation
<p>Given the equation \(4x + 2 = 2(2x + 3)\)</p> <p>First, expand the right-hand side: \(4x + 2 = 4x + 6\)</p> <p>Next, subtract \(4x\) from both sides of the equation: \(4x - 4x + 2 = 4x - 4x + 6\)</p> <p>This simplifies to: \(2 = 6\)</p> <p>Since this result is not possible (\(2\) does not equal \(6\)), the equation has no solution.</p>
CamTutor
In regards to math, we are professionals.
Get In Touch
Email: camtutor.ai@gmail.com