Find The { Y$} − I N T E R C E P T A N D T H E S L O P E O F T H E L I N E G I V E N B Y T H E E Q U A T I O N : -intercept And The Slope Of The Line Given By The Equation: − In T Erce Pt An D T H Es L O P Eo F T H E L In E G I V E Nb Y T H Ee Q U A T I O N : {3x - 4y = -20\} Write Your Answers In Simplest Form. { Y$}$-Intercept: { \square$}$Slope: { \square$}$
Introduction
In mathematics, a linear equation is a type of equation in which the highest power of the variable(s) is 1. The general form of a linear equation is ax + by = c, where a, b, and c are constants, and x and y are variables. In this article, we will focus on finding the y-intercept and slope of a line given by the equation 3x - 4y = -20.
What is the y-intercept?
The y-intercept of a line is the point at which the line intersects the y-axis. It is the value of y when x is equal to 0. In other words, it is the point on the line where the x-coordinate is 0.
Finding the y-intercept
To find the y-intercept, we need to isolate the variable y in the equation. We can do this by rearranging the equation to solve for y. Let's start by adding 4y to both sides of the equation:
3x - 4y + 4y = -20 + 4y
This simplifies to:
3x = -20 + 4y
Next, we can add 20 to both sides of the equation:
3x + 20 = 4y
Now, we can divide both sides of the equation by 4:
(3x + 20) / 4 = y
This simplifies to:
y = (3x + 20) / 4
Now that we have isolated the variable y, we can find the y-intercept by substituting x = 0 into the equation:
y = (3(0) + 20) / 4
y = (0 + 20) / 4
y = 20 / 4
y = 5
Therefore, the y-intercept of the line is 5.
What is the slope?
The slope of a line is a measure of how steep the line is. It is calculated as the ratio of the vertical change (rise) to the horizontal change (run) between two points on the line. In other words, it is the rate at which the line rises or falls as we move along the line.
Finding the slope
To find the slope, we need to rearrange the equation to isolate the variable y. We can do this by subtracting 3x from both sides of the equation:
-4y = -20 - 3x
Next, we can divide both sides of the equation by -4:
y = (-20 - 3x) / -4
This simplifies to:
y = (20 + 3x) / 4
Now that we have isolated the variable y, we can find the slope by comparing the coefficients of x and y. The coefficient of x is 3/4, and the coefficient of y is 1/4. Therefore, the slope of the line is 3/4.
Conclusion
In this article, we have learned how to find the y-intercept and slope of a line given by the equation 3x - 4y = -20. We have seen that the y-intercept is the point at which the line intersects the y-axis, and that it can be found by substituting x = 0 into the equation. We have also seen that the slope of a line is a measure of how steep the line is, and that it can be found by comparing the coefficients of x and y in the equation.
Final Answer
- y-intercept: 5
- Slope: 3/4
Additional Resources
For more information on linear equations and how to find the y-intercept and slope, please see the following resources:
- Khan Academy: Linear Equations
- Mathway: Linear Equations
- Wolfram Alpha: Linear Equations
Frequently Asked Questions
Q: What is the y-intercept of a line? A: The y-intercept of a line is the point at which the line intersects the y-axis.
Q: How do I find the y-intercept of a line? A: To find the y-intercept, substitute x = 0 into the equation and solve for y.
Q: What is the slope of a line? A: The slope of a line is a measure of how steep the line is.
Q: What is a linear equation?
A: A linear equation is a type of equation in which the highest power of the variable(s) is 1. The general form of a linear equation is ax + by = c, where a, b, and c are constants, and x and y are variables.
Q: How do I write a linear equation in slope-intercept form?
A: To write a linear equation in slope-intercept form, you need to isolate the variable y. This can be done by subtracting ax from both sides of the equation and then dividing both sides by b.
Q: What is the slope-intercept form of a linear equation?
A: The slope-intercept form of a linear equation is y = mx + b, where m is the slope and b is the y-intercept.
Q: How do I find the slope of a line?
A: To find the slope of a line, you need to compare the coefficients of x and y in the equation. The coefficient of x is the slope.
Q: What is the y-intercept of a line?
A: The y-intercept of a line is the point at which the line intersects the y-axis. It is the value of y when x is equal to 0.
Q: How do I find the y-intercept of a line?
A: To find the y-intercept, substitute x = 0 into the equation and solve for y.
Q: What is the difference between a linear equation and a quadratic equation?
A: A linear equation is a type of equation in which the highest power of the variable(s) is 1, while a quadratic equation is a type of equation in which the highest power of the variable(s) is 2.
Q: How do I solve a linear equation?
A: To solve a linear equation, you need to isolate the variable(s) by performing the necessary operations to get the variable(s) on one side of the equation.
Q: What is the order of operations for solving linear equations?
A: The order of operations for solving linear equations is:
- Parentheses: Evaluate any expressions inside parentheses.
- Exponents: Evaluate any exponential expressions.
- Multiplication and Division: Evaluate any multiplication and division operations from left to right.
- Addition and Subtraction: Evaluate any addition and subtraction operations from left to right.
Q: How do I graph a linear equation?
A: To graph a linear equation, you need to find two points on the line and then draw a line through those points.
Q: What is the significance of the slope of a line?
A: The slope of a line represents the rate of change of the line. It can be used to determine the steepness of the line.
Q: How do I find the equation of a line given two points?
A: To find the equation of a line given two points, you need to find the slope of the line and then use one of the points to find the y-intercept.
Q: What is the equation of a line in point-slope form?
A: The equation of a line in point-slope form is y - y1 = m(x - x1), where (x1, y1) is a point on the line and m is the slope.
Q: How do I convert a linear equation from standard form to slope-intercept form?
A: To convert a linear equation from standard form to slope-intercept form, you need to isolate the variable y by subtracting ax from both sides of the equation and then dividing both sides by b.
Q: What is the difference between a linear equation and a nonlinear equation?
A: A linear equation is a type of equation in which the highest power of the variable(s) is 1, while a nonlinear equation is a type of equation in which the highest power of the variable(s) is greater than 1.
Q: How do I solve a nonlinear equation?
A: To solve a nonlinear equation, you need to use numerical methods or algebraic techniques to find the solution.
Q: What is the significance of the y-intercept of a line?
A: The y-intercept of a line represents the point at which the line intersects the y-axis. It can be used to determine the value of y when x is equal to 0.
Q: How do I find the equation of a line given the slope and a point?
A: To find the equation of a line given the slope and a point, you need to use the point-slope form of a linear equation.
Q: What is the equation of a line in standard form?
A: The equation of a line in standard form is ax + by = c, where a, b, and c are constants, and x and y are variables.
Q: How do I convert a linear equation from slope-intercept form to standard form?
A: To convert a linear equation from slope-intercept form to standard form, you need to multiply both sides of the equation by the denominator and then simplify.
Q: What is the difference between a linear equation and a rational equation?
A: A linear equation is a type of equation in which the highest power of the variable(s) is 1, while a rational equation is a type of equation in which the variable(s) are in the numerator or denominator of a fraction.
Q: How do I solve a rational equation?
A: To solve a rational equation, you need to find a common denominator and then multiply both sides of the equation by the denominator.
Q: What is the significance of the slope of a line in a rational equation?
A: The slope of a line in a rational equation represents the rate of change of the line. It can be used to determine the steepness of the line.
Q: How do I find the equation of a line given the slope and a point in a rational equation?
A: To find the equation of a line given the slope and a point in a rational equation, you need to use the point-slope form of a rational equation.
Q: What is the equation of a line in a rational equation?
A: The equation of a line in a rational equation is y = mx + b, where m is the slope and b is the y-intercept.
Q: How do I convert a rational equation from standard form to slope-intercept form?
A: To convert a rational equation from standard form to slope-intercept form, you need to multiply both sides of the equation by the denominator and then simplify.
Q: What is the difference between a linear equation and a polynomial equation?
A: A linear equation is a type of equation in which the highest power of the variable(s) is 1, while a polynomial equation is a type of equation in which the highest power of the variable(s) is greater than 1.
Q: How do I solve a polynomial equation?
A: To solve a polynomial equation, you need to use numerical methods or algebraic techniques to find the solution.
Q: What is the significance of the y-intercept of a line in a polynomial equation?
A: The y-intercept of a line in a polynomial equation represents the point at which the line intersects the y-axis. It can be used to determine the value of y when x is equal to 0.
Q: How do I find the equation of a line given the slope and a point in a polynomial equation?
A: To find the equation of a line given the slope and a point in a polynomial equation, you need to use the point-slope form of a polynomial equation.
Q: What is the equation of a line in a polynomial equation?
A: The equation of a line in a polynomial equation is y = mx + b, where m is the slope and b is the y-intercept.
Q: How do I convert a polynomial equation from standard form to slope-intercept form?
A: To convert a polynomial equation from standard form to slope-intercept form, you need to multiply both sides of the equation by the denominator and then simplify.
Q: What is the difference between a linear equation and a trigonometric equation?
A: A linear equation is a type of equation in which the highest power of the variable(s) is 1, while a trigonometric equation is a type of equation in which the variable(s) are trigonometric functions.
Q: How do I solve a trigonometric equation?
A: To solve a trigonometric equation, you need to use trigonometric identities and algebraic techniques to find the solution.
Q: What is the significance of the y-intercept of a line in a trigonometric equation?
A: The y-intercept of a line in a trigonometric equation represents the point at which the line intersects the y-axis. It can be used to determine the value of y when x is equal to 0.
Q: How do I find the equation of a line given the slope and a point in a trigonometric equation?
A: To find the equation of a line given the slope and a point in a trigonometric equation, you need to use the point-slope form of a trigonometric equation.