What does a linear EQUATION look like?
y = mx + b
What does a linear GRAPH look like?
A line. Lines have a constant slope.
What does a linear TABLE or LIST look like?
There is a common difference between outputs. The average rate of change NEVER varies, no matter which two points you use for A.R.C.
What does linear CONTEXT look like?
Linear context has some starting point, or base amount, and increases or decreases by a fixed numerical amount.
Slope
Average Rate of Change. Also known as rise over run.
Three forms for linear functions
- Slope - Intercept
- Point-Slope
- Standard
Point - Slope Form
y - k = m(x-h)
(h,k) is a point, m is the slope.
Standard Form
Ax + By = C
List out the steps for turning standard form into slope-intercept form
Ax + By = C
- Subtract Ax from both sides: By = C - Ax
- Divide both sides by B: y = C/B - Ax/B
How do you find x-intercept in standard form?
Ax + By = C
- Substitute y = 0. So, Ax = C.
- Divide by A. So, x= C/A
How do you find a y-intercept in Standard Form?
Ax + By = C
- Substitute x = 0. So, By = C
- Divide both sides by B: y = C/B
y = mx + b
What do m and b represent?
m = slope b = y-intercept
What is a system of equations?
A system of equations is a set of two equations (with an input x and an output y) where you find the common solution point(s): (x,y)
What are the four methods for solving a system?
- Substitution
- Elimination
- Graphing
- Guess and Check
y = 3x + 5 y = 4x-7
How should you solve this system?
- Substitution: 4x - 7 = 3x + 5
or. . - Graphing: Find the intersection of the two lines.
3x - 4y = 10
2x + y = 5
How should you solve the system?
Elimination.
If you have desmos, use graphing.
3x - 4y = 10
2x + y = 5
How would you begin using elimination?
Multiply equation #2 by four
5x - y = 10
x = 3y + 2
How would you solve the system?
Substitute the second equation into the first: 5(3y+2) - y = 10.
Describe systems of linear inequalities.
Graphs of two intersecting lines (solid or dotted) where the solutions are represented with intersecting shaded regions.
What is the only method to solve systems of linear inequalities?
Graphing (by hand or with desmos).
y > 3x + 5
How do you graph this by hand?
- Graph y = 3x + 5 but make it dotted due to > (has no equality)
- Shade above the line (because of greater than)
- For a system, just do this process twice. Intersecting shades are solution.
How can you verify if a point is a solution to a system of inequalities?
Two different ways:
- Visually inspect. Check that the point is in the DOUBLE SHADED region.
- Evaluate (x,y) into BOTH inequalities. If both are true, then (x,y) is a solution.
How do you graph 3x + 2y <= 5?
- Go from standard form to slope-intercept form: 2y <= 5 - 3x……. y <= 2.5 - 3/2 x
- Make sure line is SOLID
- Shade below the line
Is (5, 10) in the solution region for:
y > 3x - 10
2x + 3y < 6
?
No.
- 10 > 3(5) - 10. So 10 > 5
- 2(5) + 3(10) < 6 ? So 40 < 6. False.
