Use the given conditions to write an equation for the line in point-slope form and slope-intercept form.
Passing through $(-5,-5)$ and $(5,7)$
Type the point-slope form of the equation of the line.
[tex]y+5=\frac{6}{5}(x+5)[/tex]
(Use integers or simplified fractions for any numbers in the equation.)
Type the slope-intercept form of the equation of the line.
[tex]\square[/tex]
(Use integers or simplified fractions for any numbers in the equation.)
Answer (2)
Distribute the slope in the point-slope form: y + 5 = 5 6 ( x + 5 ) becomes y + 5 = 5 6 x + 6 .
Isolate y by subtracting 5 from both sides: y = 5 6 x + 6 − 5 .
Simplify the equation: y = 5 6 x + 1 .
The slope-intercept form of the equation is: y = 5 6 x + 1 .
Explanation
Understanding the Problem We are given a line that passes through the points ( − 5 , − 5 ) and ( 5 , 7 ) . We are also given the point-slope form of the equation of the line as y + 5 = 5 6 ( x + 5 ) . Our goal is to find the slope-intercept form of the equation of the line.
Understanding the Forms The point-slope form of a line is given by y − y 1 = m ( x − x 1 ) , where ( x 1 , y 1 ) is a point on the line and m is the slope of the line. The slope-intercept form of a line is given by y = m x + b , where m is the slope and b is the y-intercept. We are given the point-slope form y + 5 = 5 6 ( x + 5 ) . We want to convert this to the slope-intercept form.
Starting with Point-Slope Form To convert the point-slope form to slope-intercept form, we need to isolate y on the left side of the equation. We start with the given point-slope form:
y + 5 = 5 6 ( x + 5 )
Distributing the Slope Next, we distribute the 5 6 on the right side of the equation:
y + 5 = 5 6 x + 5 6 ( 5 )
Simplifying the Equation Now, we simplify the right side:
y + 5 = 5 6 x + 6
Isolating y To isolate y , we subtract 5 from both sides of the equation:
y = 5 6 x + 6 − 5
Slope-Intercept Form Finally, we simplify to get the slope-intercept form:
y = 5 6 x + 1
So, the slope-intercept form of the equation of the line is y = 5 6 x + 1 .
Examples
Understanding linear equations is crucial in many real-world applications. For instance, imagine you're tracking the growth of a plant. If the plant grows at a constant rate, say 5 6 inches per day, and it started at a height of 1 inch, the equation y = 5 6 x + 1 models its growth. Here, y represents the height of the plant after x days. This kind of linear relationship is also applicable in calculating simple interest, where the amount of interest earned is directly proportional to the principal amount and the interest rate.
The point-slope form of the line is y + 5 = 5 6 ( x + 5 ) . The slope-intercept form is y = 5 6 x + 1 .
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