14.Use the information given to determine the equation of each line.
a. the slope of the line is 4/3 and its y-intercept is -2
b.the line perpendicular to the line y=-2x+5 and passing through the point (-2,6)
14a. Y=mX+b, m=slope, b=Y-intercept.
Equation:Y=4/3X-2
14b. Given: Y=-2X+5, p(-2,6). m1=-2
m2=the negative reciprocal of m1:
m2=1/2, Y=mx+b, 6=1/2(-2)+b, b=7,
Equation: Y=1/2X+7.
a. To determine the equation of a line when given its slope and y-intercept, you can use the slope-intercept form of a linear equation, which is y = mx + b. The slope (m) represents the rate of change of the line, while the y-intercept (b) represents the y-coordinate where the line intersects the y-axis.
For this question, the slope is 4/3 and the y-intercept is -2. Substituting these values into the slope-intercept form, we get:
y = (4/3)x - 2
Therefore, the equation of the line is y = (4/3)x - 2.
b. To find the equation of a line perpendicular to a given line, we need to find the negative reciprocal of the slope of the given line. The negative reciprocal is found by taking the negative value of the reciprocal.
Given the line y = -2x + 5, we can determine its slope by comparing it to the standard slope-intercept form of a linear equation. The slope is the coefficient of x, which in this case is -2.
To find the negative reciprocal of -2, we take the reciprocal of -2 and then change its sign. The reciprocal of -2 is -1/2, so the negative reciprocal is 1/2.
We also know that the line passes through the point (-2,6). Using this point and the negative reciprocal of the slope, we can determine the equation using the point-slope form of a linear equation, which is y - y1 = m(x - x1).
Substituting the values into the equation, we have:
y - 6 = (1/2)(x - (-2))
Simplifying further:
y - 6 = (1/2)(x + 2)
y - 6 = (1/2)x + 1
y = (1/2)x + 7
Therefore, the equation of the line perpendicular to y = -2x + 5 and passing through the point (-2,6) is y = (1/2)x + 7.