Example Question - slope-intercept form
Here are examples of questions we've helped users solve.
Graphing Linear Equations
<p>Para graficar la ecuación lineal \( y = 3x - 2 \), primero evaluamos \( y \) para cada valor de \( x \) en la tabla proporcionada.</p> <p>Si \( x = 3 \):</p> <p>\( y = 3(3) - 2 = 9 - 2 = 7 \)</p> <p>Si \( x = 2 \):</p> <p>\( y = 3(2) - 2 = 6 - 2 = 4 \)</p> <p>Si \( x = -2 \):</p> <p>\( y = 3(-2) - 2 = -6 - 2 = -8 \)</p> <p>Y cuando \( x = 0 \), que ya está dado en la tabla:</p> <p>\( y = 3(0) - 2 = 0 - 2 = -2 \)</p> <p>Ahora podemos graficar los puntos (3,7), (2,4), (-2,-8), y (0,-2) en el plano coordenado y dibujar la línea que los une, que será la gráfica de la ecuación lineal \( y = 3x - 2 \).</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 \( y = 3x - 2 \).
Finding the Equation of a Line Given Slope and a Point
Given slope, \( m = 2 \), and a point, \( (1,3) \), use the point-slope form of the equation of a line: \( y - y_1 = m(x - x_1) \). Substitute \( m = 2 \), \( x_1 = 1 \), and \( y_1 = 3 \) into the equation: \( y - 3 = 2(x - 1) \) Now, simplify and put it in slope-intercept form, \( y = mx + b \): \( y - 3 = 2x - 2 \) \( y = 2x + 1 \)
Explanation of Slope-Intercept Form
The given equation is the slope-intercept form of a line, which is represented as \( y = mx + b \), where: - \( m \) is the slope of the line - \( b \) is the y-intercept, i.e., the value of \( y \) when \( x = 0 \) There is no specific question to solve, as the image simply provides the general form of a linear equation in slope-intercept form.
Finding the Equation of a Line Given Slope and a Point
Given, slope \( m = \frac{2}{3} \) and point \( (1, -2) \). The equation of a line in slope-intercept form is \( y = mx + b \). Substituting the given slope and point into the equation to find \( b \): \( -2 = \left( \frac{2}{3} \right)(1) + b \) To solve for \( b \): \( b = -2 - \frac{2}{3} \) \( b = -\frac{6}{3} - \frac{2}{3} \) \( b = -\frac{8}{3} \) The equation of the line is: \( y = \frac{2}{3}x - \frac{8}{3} \)
Solving System of Linear Equations by Graphing with Same Slope
To solve a system of linear equations by graphing, we need to plot each equation on a graph and identify where they intersect. The equations given are: 1) x + y = 2 2) x + y = 3 For each equation, we can solve for y to put the equation into slope-intercept form (y = mx + b). For the first equation: x + y = 2 y = -x + 2 (subtracting x from both sides) For the second equation: x + y = 3 y = -x + 3 (subtracting x from both sides) Now let's graph each equation. We'll start by plotting the y-intercept for each line, which is the point where x = 0. For the first equation (y = -x + 2), when x = 0, y = 2, so we have the point (0, 2). For the second equation (y = -x + 3), when x = 0, y = 3, so we have the point (0, 3). Next, we use the slope, which is -1 (since the coefficient of x is -1 in both equations), to find another point for each line. From our y-intercept (0, 2) for the first equation, move down 1 unit and to the right 1 unit to reach the point (1, 1). Similarly, from the y-intercept (0, 3) for the second equation, move down 1 unit and to the right 1 unit to reach the point (1, 2). Now, we can draw the lines through these points: For the first line: Draw a line through (0, 2) and (1, 1) For the second line: Draw a line through (0, 3) and (1, 2) However, upon examining these two equations, we realize they are parallel and have the same slope but different y-intercepts. Since they are parallel, they will never intersect. This means there is no solution to this system of equations; it is inconsistent. The graph provided in the image seems to be showing the lines incorrectly as they intersect, but that must be a mistake because, theoretically, the two lines cannot intersect given they have the same slope and different y-intercepts.
Converting Function to Slope-Intercept Form
The function given in the image is: \[ f(x) = -2 \left( x + \frac{1}{4} \right) - \frac{49}{8} \] To express the function in the form of y-intercept, we need to simplify it while keeping it in the form of \( y = mx + b \), where \( m \) stands for the slope and \( b \) is the y-intercept. Let's simplify the function: \[ f(x) = -2x - \frac{1}{2} - \frac{49}{8} \] To combine the constant terms, we need a common denominator. The common denominator for \( \frac{1}{2} \) and \( \frac{49}{8} \) is 8. So let's convert \( \frac{1}{2} \) to have the denominator of 8: \[ - \frac{1}{2} = - \frac{4}{8} \] Now rewritting the equation we have: \[ f(x) = -2x - \frac{4}{8} - \frac{49}{8} = -2x - \frac{53}{8} \] So the function in the form of intercept is: \[ f(x) = -2x - \frac{53}{8} \] This function is now in slope-intercept form with \( m = -2 \) (the slope) and \( b = -\frac{53}{8} \) (the y-intercept).
Equation of Perpendicular Line
The equation of line p is given as y = -4x + 1. This equation is in slope-intercept form, y = mx + b, where m is the slope and b is the y-intercept. For line p, the slope is -4. Line q is supposed to be perpendicular to line p. When two lines are perpendicular, the product of their slopes is -1. This means that if the slope of line p is m, then the slope of line q will be -1/m. Therefore, the slope of line q is -1/(-4) = 1/4. Now that we know the slope of line q is 1/4, we can use the point it passes through, (-6, 1), to find the y-intercept (b) of line q. Starting with the point-slope form of the line equation: y - y1 = m(x - x1) Plugging in the slope (m = 1/4) and the point (-6, 1): y - 1 = 1/4(x - (-6)) y - 1 = 1/4(x + 6) Now, distribute 1/4 to (x + 6): y - 1 = 1/4x + 1/4(6) Simplify: y - 1 = 1/4x + 6/4 y - 1 = 1/4x + 3/2 Finally, we want to write this in slope-intercept form, so we solve for y by adding 1 to both sides: y = 1/4x + 3/2 + 1 Since we want to write numbers as simplified fractions or integers, let's convert 1 to a fraction with a denominator of 2: y = 1/4x + 3/2 + 2/2 Combine the fractions: y = 1/4x + 5/2 And so, the equation of line q in slope-intercept form is: y = 1/4x + 5/2
Finding Equation of Perpendicular Line
To find the equation of line h, which is perpendicular to line g, we will follow these steps: 1. Identify the slope of line g. 2. Determine the slope of line h. 3. Use the point-slope form to create the equation of line h. 4. Convert the equation into slope-intercept form. The equation for line g is given as \( y = \frac{3}{10}x - 8 \). The slope of line g is the coefficient of x, which is \( \frac{3}{10} \). Lines that are perpendicular to each other have slopes that are negative reciprocals. Therefore, if the slope of line g is \( \frac{3}{10} \), the slope of line h will be its negative reciprocal, which is \( -\frac{10}{3} \). Now, we know that line h has a slope of \( -\frac{10}{3} \) and it passes through the point (3, -9). We can use the point-slope form of the equation to find the equation of line h: \( y - y_1 = m(x - x_1) \) Substitute m (slope) with \( -\frac{10}{3} \) and \( (x_1, y_1) \) with (3, -9): \( y - (-9) = -\frac{10}{3}(x - 3) \) Simplify and solve for y to get the equation in slope-intercept form: \( y + 9 = -\frac{10}{3}x + 10 \) \( y = -\frac{10}{3}x + 10 - 9 \) \( y = -\frac{10}{3}x + 1 \) Therefore, the equation of line h in slope-intercept form is \( y = -\frac{10}{3}x + 1 \), with the numbers in the equation as simplified improper fractions or integers.
Finding a Parallel Line Passing Through a Point
The equation for line q is given as: \( y = -5 - \frac{1}{8}(x + 2) \) To find a line parallel to q that passes through the point (-6, 1), we need to keep the slope the same, since parallel lines have equal slopes. First, let's rewrite the equation for line q in slope-intercept form (\( y = mx + b \)), where \( m \) is the slope and \( b \) is the y-intercept. Rewrite the equation of line q to make the slope more apparent: \( y = -5 - \frac{1}{8}x - \frac{1}{8}(2) \) \( y = -5 - \frac{1}{8}x - \frac{1}{4} \) \( y = -\frac{1}{8}x - 5 - \frac{1}{4} \) \( y = -\frac{1}{8}x - 5.25 \) \( y = -\frac{1}{8}x - \frac{21}{4} \) Now we know the slope of line q is \( -\frac{1}{8} \). Since line r is parallel to line q, it will also have a slope of \( -\frac{1}{8} \). Using the point-slope form of a line's equation ( \( y - y_1 = m(x - x_1) \) ), where \( m \) is the slope and \( (x_1, y_1) \) is the point (-6, 1) through which line r passes, we can write: \( y - 1 = -\frac{1}{8}(x - (-6)) \) \( y - 1 = -\frac{1}{8}(x + 6) \) Now we can put this in slope-intercept form: \( y = -\frac{1}{8}x - \frac{1}{8}(6) + 1 \) \( y = -\frac{1}{8}x - \frac{3}{4} + 1 \) \( y = -\frac{1}{8}x + \frac{1}{4} \) So, the equation of line r in slope-intercept form is: \( y = -\frac{1}{8}x + \frac{1}{4} \)
Finding Equation of Parallel Line Passing Through a Point
To find the equation of line r that is parallel to line q and passes through the point (-6, 1), we start by determining the slope of line q. The equation of line q is given in slope-intercept form as: y = -5 - 1/2(x + 2) In slope-intercept form, which is y = mx + b, m represents the slope and b represents the y-intercept. Based on line q's equation, the slope (m) is -1/2. Since line r is parallel to line q, line r will have the same slope as line q. Therefore, the slope of line r will also be -1/2. Using the point-slope form of a line equation, which is y - y1 = m(x - x1), where (x1, y1) is a point on the line, and m is the slope, we can substitute the slope and the point through which line r passes. The point (-6, 1) will be our (x1, y1), and our slope (m) will be -1/2. y - y1 = m(x - x1) y - 1 = -1/2(x - (-6)) y - 1 = -1/2(x + 6) Now, we solve for y to get the equation in slope-intercept form. y = -1/2 * x - 1/2 * 6 + 1 y = -1/2 * x - 3 + 1 y = -1/2 * x - 2 So, the equation of line r in slope-intercept form is: y = -1/2x - 2 This equation uses numbers written as simplified proper fractions, improper fractions, or integers, as requested.
Solving Linear Equations in Slope-Intercept Form
The equation in the image is a linear equation in slope-intercept form, which is generally expressed as y = mx + b, where m is the slope and b is the y-intercept. To solve the equation y - 1 = -3(x - 5), we will simplify and put it in slope-intercept form. Here are the steps: 1. Distribute -3 into the parentheses: y - 1 = -3 * x + 3 * 5 y - 1 = -3x + 15 2. Add 1 to both sides to isolate y: y = -3x + 15 + 1 y = -3x + 16 Now the equation is in slope-intercept form y = -3x + 16, where the slope (m) is -3 and the y-intercept (b) is 16.
Converting Equation to Slope-Intercept Form and Graphing
The equation provided in the image appears to be in the form: \[ y + 6 = -\frac{3}{4}(x + 3) \] To graph this line, we need to bring it into slope-intercept form, which is \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept. We need to isolate \( y \) on one side of the equation: \[ y = -\frac{3}{4}(x + 3) - 6 \] To get rid of the parentheses, distribute \( -\frac{3}{4} \): \[ y = -\frac{3}{4}x - \frac{3}{4} \cdot 3 - 6 \] Calculating \( -\frac{3}{4} \cdot 3 \): \[ y = -\frac{3}{4}x - \frac{9}{4} - 6 \] We can convert 6 into fraction form to combine it with \( -\frac{9}{4} \): \[ -6 = -\frac{24}{4} \] So now we have: \[ y = -\frac{3}{4}x - \frac{9}{4} - \frac{24}{4} \] Combine the fractions: \[ y = -\frac{3}{4}x - \frac{33}{4} \] Now the equation is in slope-intercept form, where the slope \( m \) is \( -\frac{3}{4} \) and the y-intercept \( b \) is \( -\frac{33}{4} \). To graph this line: - Start by plotting the y-intercept at \( 0, -\frac{33}{4} \) (or if you prefer a decimal, approximately \( 0, -8.25 \)). - From the y-intercept, use the slope to determine the direction and steepness of the line. The slope \( -\frac{3}{4} \) means that for every 4 units you move to the right (positive direction along the x-axis), you move 3 units down (negative direction along the y-axis), because the slope is negative. - Plot another point using the slope from the y-intercept and then draw the line through both points extending it across the graph.
Graphing Equations in Slope-Intercept Form
To graph the equation \( y + 6 = -\frac{3}{4}(x + 3) \), you should first rewrite it in slope-intercept form, which is \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept. Starting with the given equation: \( y + 6 = -\frac{3}{4}(x + 3) \) Isolate \( y \) on one side of the equation: \( y = -\frac{3}{4}(x + 3) - 6 \) Now distribute \( -\frac{3}{4} \) to both terms inside the parentheses: \( y = -\frac{3}{4}x - \frac{3}{4} \cdot 3 - 6 \) Multiply \( -\frac{3}{4} \) by 3 to get \( -\frac{9}{4} \), and then continue to simplify: \( y = -\frac{3}{4}x - \frac{9}{4} - 6 \) Since \( 6 = \frac{24}{4} \), we have: \( y = -\frac{3}{4}x - \frac{9}{4} - \frac{24}{4} \) Combine the constant terms: \( y = -\frac{3}{4}x - \frac{33}{4} \) So the equation in slope-intercept form is: \( y = -\frac{3}{4}x - \frac{33}{4} \) Now you can graph the line with the slope \( -\frac{3}{4} \) and a y-intercept \( -\frac{33}{4} \). Start by plotting the y-intercept on the y-axis at the point (0, -\( \frac{33}{4} \)) or (0, -8.25). Then, use the slope to find another point. From the y-intercept, move 3 units down and 4 units to the right (since the slope is negative), which will give you another point on the line. Connect these points with a straight line, and you will have graphed the given equation.
Graphing a Line in Slope-Intercept Form
To graph the line given by the equation: \[ y + 3 = \frac{9}{4}(x + 4) \] We should first rewrite it into the slope-intercept form, which is \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept. Let's solve the equation for \( y \): \[ y = \frac{9}{4}(x + 4) - 3 \] Now, distribute the \( \frac{9}{4} \) across \( (x + 4) \): \[ y = \frac{9}{4}x + \frac{9}{4} \cdot 4 - 3 \] Simplify the constants: \[ y = \frac{9}{4}x + 9 - 3 \] \[ y = \frac{9}{4}x + 6 \] Now we have the equation in slope-intercept form with a slope \( m = \frac{9}{4} \) and a y-intercept \( b = 6 \). To graph this line, follow these steps: 1. Start by plotting the y-intercept (0,6) on the graph. 2. From this point, use the slope to determine the next point. The slope is \( \frac{9}{4} \) which means that for every 4 units you move to the right (in the positive x-direction), you move 9 units up (in the positive y-direction). 3. Plot another point using the slope. For example, starting at (0,6), go right 4 units to (4,6) and then up 9 units to (4,15). 4. Draw a straight line through the points to complete the graph of the line. Remember, you can plot more points if needed to ensure accuracy before drawing your line.
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