slope
Given the equation of a line y=3x+5 what is the slope of the line?
What is the slope?
The slope is 3, because the equation is in the for y=mx + b, where m =3.
In the equation of a line y=mx +b, what does b represent?
The y-intercept is the value of y where the line crosses the y-axis. It occurs when x=0,
Solve the system of equations:
3x+2y=12
4x−y=5
Use substitution or elimination.
From the second equation, solve for y.
y=4x−5
Substitute into the first equation
3x+2(4x−5)=12
3x+8x−10=12
11x=22
x=2
Substitute x=2 into y=4x−5
y=4(2)−5=3.
The solution is (2,3).
1. Solve the second equation for x:
x = y + 2
2. Substitute x = y + 2 into the first equation:
3(y + 2) + 2y = 18
3y + 6 + 2y = 18
5y + 6 = 18
5y = 12
y = \frac{12}{5}
3. Substitute y = \frac{12}{5} into x = y + 2:
x = \frac{12}{5} + 2 = \frac{12}{5} + \frac{10}{5} = \frac{22}{5}
x = \frac{22}{5}, y = \frac{12}{5}
Problem 1:
y = 2x + 1
y = -x + 4
Solution:
The point of intersection occurs where 2x + 1 = -x + 4:
2x + x = 4 - 1
3x = 3 \implies x = 1
Substitute x = 1 into y = 2x + 1:
y = 2(1) + 1 = 3
C
(1, 3)
This is the formula used to calculate the slope between two points, (x1,y1)and (x2,y2).
The slope formula is:
m= y2-y1
x2-x1
What is the y-intercept of the equation 3x−4y=12?
To find the y-intercept, set x=0 in the equation.
3(0)−4y=12
−4y=12
y=−3.
Solve the system of equations:
x−2y=4
3x+y=7
From the first equation, solve for x
x=2y+4
Substitute into the second equation:
3(2y+4)+y=7
6y+12+y=7
7y=−5
Substitute y=−5/7 into x=2y+4
x=2(−5/7)+4=18/7.
The solution is
(18/7,−5/7)
Solve the first equation for y:
y = 10 - x
2. Substitute y = 10 - x into the second equation:
2x - (10 - x) = 4
2x - 10 + x = 4
3x = 14
x = \frac{14}{3}
3. Substitute x = \frac{14}{3} back into y = 10 - x:
y = 10 - \frac{14}{3} = \frac{30}{3} - \frac{14}{3} = \frac{16}{3}
x = \frac{14}{3}, y = \frac{16}{3}
y = \frac{1}{2}x - 3
y = -\frac{2}{3}x + 2
\left(\frac{30}{7}, -\frac{6}{7}\right)
If a line passes through the points (1,2 and (4,−1) what is the slope of the line?
Using the slope formula
m=y2−y1=4 =−1−1−2= -3
x2−x1y2−y1 = 4−1−1−2=3−3=−1
so,the slope is -1
The line with equation 2x+5y=20 crosses the y-axis at which point?
Set x=0 to find the y-intercept.
2(0)+5y=20
5y=20
y=4.
The y-intercept is (0,4)
Solve the system of equations:
5x+3y=11
−2x+y=−1
From the second equation, solve for y
Substitute into the first equation:
5x+3(2x−1)=11
5x+6x−3=11
11x=14
x=14/11
Substitute
x=14/11into y=2x−1y=2x−1
y=2(14/11)−1=17/11.
The solution is (14/11,17/11)
This is the point where the graphs of two equations intersect, representing the solution to a system of equations.
point of intersection
This graph of a quadratic function takes the shape of a U and is described by the equation y = ax^2 + bx + c . What is it called?
parabola
Two lines are parallel. If one line has the equation y=2x−4, what is the slope of the other line?
Parallel lines have the same slope. The slope of the given line is 2, so the slope of the parallel line is also 2.
If the equation of a line is 4x−7y=28, what is the y-intercept?
Set x=0 in the equation.
4(0)−7y=28
−7y=28
y=−4
The y-intercept is (0,−4)
This type of equation has a variable raised to the first power and graphs as a straight line.
linear equation
This method involves adding or subtracting two equations to eliminate one variable in a system of equations.
elimination
What is the equation of a line with a slope of -3 and a y-intercept of 5 ?
y = -3x + 5
A car travels 150 miles east and 50 miles north. If the car's starting point is at the origin, what is the slope of the path the car took?
The slope is the ratio of the change in y (northward distance) to the change in x (eastward distance).
m= 50/150 =1/3
so, the slope is 1/3
Find the y-intercept of the line given by 6x+3y=−9
Set x=0to find the y-intercept.
6(0)+3y=−9
3y=−9
y=−3
The y-intercept is (0,−3)
This is the property that states you can add, subtract, multiply, or divide the same value to both sides of an equation without changing the solution
Equality Property of Equations
This technique involves substituting the expression for one variable from one equation into the other equation in a system.
substitution
This is the point where a line crosses the y-axis.
Y intercept