Find A Point On The Line And The Line's Slope.Equation: Y − 4 = − 1 2 ( X − 3 Y - 4 = -\frac{1}{2}(x - 3 Y − 4 = − 2 1 ( X − 3 ]- Point On The Line: ( ⟦ □ , □ (\llbracket \square, \square ( [ [ □ , □ ]- Slope: □ \square □
Introduction
In mathematics, a line is a set of points that extend infinitely in two directions. It is often represented by an equation in the form of y = mx + b, where m is the slope and b is the y-intercept. In this article, we will discuss how to find a point on a line and the line's slope given an equation in the form of y - 4 = -\frac{1}{2}(x - 3).
Understanding the Equation
The given equation is y - 4 = -\frac{1}{2}(x - 3). To find a point on the line, we need to isolate y. We can do this by adding 4 to both sides of the equation.
Isolating y
y - 4 + 4 = -\frac{1}{2}(x - 3) + 4
y = -\frac{1}{2}(x - 3) + 4
Now, we can simplify the equation by distributing the negative sign to the terms inside the parentheses.
Distributing the Negative Sign
y = -\frac{1}{2}x + \frac{3}{2} + 4
y = -\frac{1}{2}x + \frac{11}{2}
Finding a Point on the Line
To find a point on the line, we need to choose a value for x and substitute it into the equation. Let's choose x = 0.
y = -\frac{1}{2}(0) + \frac{11}{2}
y = \frac{11}{2}
So, the point on the line is (0, \frac{11}{2}).
Finding the Slope
The slope of the line is the coefficient of x in the equation. In this case, the slope is -\frac{1}{2}.
Conclusion
In conclusion, we have found a point on the line and the line's slope given an equation in the form of y - 4 = -\frac{1}{2}(x - 3). The point on the line is (0, \frac{11}{2}) and the slope is -\frac{1}{2}.
Step-by-Step Solution
Step 1: Isolate y
Add 4 to both sides of the equation to isolate y.
y - 4 + 4 = -\frac{1}{2}(x - 3) + 4
y = -\frac{1}{2}(x - 3) + 4
Step 2: Distribute the Negative Sign
Distribute the negative sign to the terms inside the parentheses.
y = -\frac{1}{2}x + \frac{3}{2} + 4
y = -\frac{1}{2}x + \frac{11}{2}
Step 3: Find a Point on the Line
Choose a value for x and substitute it into the equation. Let's choose x = 0.
y = -\frac{1}{2}(0) + \frac{11}{2}
y = \frac{11}{2}
Step 4: Find the Slope
The slope of the line is the coefficient of x in the equation. In this case, the slope is -\frac{1}{2}.
Example Problems
Problem 1
Find a point on the line and the line's slope given the equation y - 2 = \frac{3}{4}(x - 1).
Solution
To find a point on the line, we need to isolate y. We can do this by adding 2 to both sides of the equation.
y - 2 + 2 = \frac{3}{4}(x - 1) + 2
y = \frac{3}{4}(x - 1) + 2
Now, we can simplify the equation by distributing the fraction to the terms inside the parentheses.
y = \frac{3}{4}x - \frac{3}{4} + 2
y = \frac{3}{4}x + \frac{5}{4}
To find a point on the line, we need to choose a value for x and substitute it into the equation. Let's choose x = 0.
y = \frac{3}{4}(0) + \frac{5}{4}
y = \frac{5}{4}
So, the point on the line is (0, \frac{5}{4}).
The slope of the line is the coefficient of x in the equation. In this case, the slope is \frac{3}{4}.
Problem 2
Find a point on the line and the line's slope given the equation y + 1 = -2(x + 2).
Solution
To find a point on the line, we need to isolate y. We can do this by subtracting 1 from both sides of the equation.
y + 1 - 1 = -2(x + 2) - 1
y = -2(x + 2) - 1
Now, we can simplify the equation by distributing the negative sign to the terms inside the parentheses.
y = -2x - 4 - 1
y = -2x - 5
To find a point on the line, we need to choose a value for x and substitute it into the equation. Let's choose x = 0.
y = -2(0) - 5
y = -5
So, the point on the line is (0, -5).
The slope of the line is the coefficient of x in the equation. In this case, the slope is -2.
Conclusion
Frequently Asked Questions
Q: What is the equation of a line in slope-intercept form?
A: The equation of a line in slope-intercept form is y = mx + b, where m is the slope and b is the y-intercept.
Q: How do I find a point on the line given an equation in slope-intercept form?
A: To find a point on the line, you need to choose a value for x and substitute it into the equation. Then, solve for y.
Q: How do I find the slope of a line given an equation in slope-intercept form?
A: The slope of a line is the coefficient of x in the equation. In other words, it is the number that is multiplied by x.
Q: What is the difference between the slope and the y-intercept?
A: The slope is the rate of change of the line, while the y-intercept is the point where the line intersects the y-axis.
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 use the point-slope form of a line, which is y - y1 = m(x - x1), where (x1, y1) and (x2, y2) are the two points.
Q: What is the point-slope form of a line?
A: The point-slope form of a line is y - y1 = m(x - x1), where (x1, y1) is a point on the line and m is the slope.
Q: How do I find the slope of a line given two points?
A: To find the slope of a line given two points, you need to use the formula m = (y2 - y1) / (x2 - x1), where (x1, y1) and (x2, y2) are the two points.
Q: What is the formula for the slope of a line given two points?
A: The formula for the slope of a line given two points is m = (y2 - y1) / (x2 - x1), where (x1, y1) and (x2, y2) are the two points.
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 line, which is y - y1 = m(x - x1), where (x1, y1) is the point and m is the slope.
Q: What is the point-slope form of a line given the slope and a point?
A: The point-slope form of a line given the slope and a point is y - y1 = m(x - x1), where (x1, y1) is the point and m is the slope.
Q: How do I find the equation of a line given the slope and the y-intercept?
A: To find the equation of a line given the slope and the y-intercept, you need to use the slope-intercept form of a line, which is y = mx + b, where m is the slope and b is the y-intercept.
Q: What is the slope-intercept form of a line given the slope and the y-intercept?
A: The slope-intercept form of a line given the slope and the y-intercept is y = mx + b, where m is the slope and b is the y-intercept.
Example Problems
Problem 1
Find the equation of a line given the slope and a point. The slope is 2 and the point is (1, 3).
Solution
To find the equation of a line given the slope and a point, we need to use the point-slope form of a line, which is y - y1 = m(x - x1), where (x1, y1) is the point and m is the slope.
y - 3 = 2(x - 1)
y - 3 = 2x - 2
y = 2x + 1
So, the equation of the line is y = 2x + 1.
Problem 2
Find the equation of a line given the slope and the y-intercept. The slope is 3 and the y-intercept is 2.
Solution
To find the equation of a line given the slope and the y-intercept, we need to use the slope-intercept form of a line, which is y = mx + b, where m is the slope and b is the y-intercept.
y = 3x + 2
So, the equation of the line is y = 3x + 2.
Conclusion
In conclusion, we have answered some frequently asked questions about finding a point on the line and the line's slope. We have also solved two example problems to demonstrate the steps involved in finding the equation of a line given the slope and a point, and the slope and the y-intercept.