1. Linear Equations
Notes & Practice

2. Things to remember
An equation = a statement in algebra that says 2 expressions are equivalent
Variables in linear equations only have power of 1
You must balance each side at all times (What you do to one side MUST be done to the other side!).

3. How to approach them
If equation has fractions, but the answers don’t, try clearing the fractions in the equation by multiplying by the least common denominator
The presence of fractions in the answer choices likely means you’ll need to rely on techniques for combining and simplifying fractions to get to the right answer
Seeing decimals in the answer choices likely indicates that using your calculator will save time on test day.

4. Working with Linear Graphs
When a linear equation in written in slope-intercept form (y= mx+b), the variable m gives the slope of the line & b represents the point at which the line intersects the y-axis
In a real-world scenario, slope represents a unit rate and the y-intercept represents a starting amount
The rate of change (slope) for a linear relationship is constant (DOES NOT VARY)
Slope is given by the formula m = y2 - y1/ x2 - x1
(X1, y1) & (x2, y2) are coordinates of points on the line
*Remember RISE OVER RUN

5. Working with Linear Graphs cont.
A line with a positive slope runs up and to the right (“uphill”)
A line with a negative slope runs down and to the right (“downhill”)
A horizontal line has a slope of 0 (with NO rise to left or right!)
A vertical line has an undefined slope
Parallel lines have the same slope
Perpendicular lines have negative reciprocal slopes (ex: 3 and -⅓)
*To find a graph that matches a given equation (& vice versa), find the slope (m) of the line and its y-intercept (b).

6. SLOPE = RATE

7. x= independent variable
y= dependent variable
“Infinite # of solutions” = solve for the variable

8. Practice 1
3y + 2(y-2) = -25
What value of y satisfies the equation above?
-29/5
-21/5
21/5
29/5

9. Practice 1 answer 3y + 2(y-2) = -25
What value of y satisfies the equation above?
-29/5
-21/5
21/5
29/5

10. Reasoning
Start by distributing the 2. Then collect like terms until you isolate y.
3y + 2(y-2) = -25
3y + 2y - 4 = -25
5y - 4 = -25
Y = -21/5

11. Line L has an undefined slope. Line M is perpendicular to line L
Line L has an undefined slope. Line M is perpendicular to line L. Which of the following could be the equation of line M?
x=y
y=7
x=-3
xy=4

12. Line L has an undefined slope. Line M is perpendicular to line L
Line L has an undefined slope. Line M is perpendicular to line L. Which of the following could be the equation of line M?
x=y
y=7
x=-3
xy=4
Reasoning: Undefined slope = VERTICAL line; therefore, parallel to y-axis!
A line that is perpendicular to a vertical line MUST be a horizontal line. Horizontal line slope = 0
Its equation, therefore, will be of the form y=0x+b
It can also be seen as y=b
Remember that b is constant. The only equation that meets this criterion is choice B!

13. ⅔ x+cy=2
If the slope of the equation above is 6, what is the value of c? -4
-1/9 ⅓ 4

14. If the slope of the equation above is 6, what is the value of c?
⅔ x+cy=2
If the slope of the equation above is 6, what is the value of c? -4
-1/9 ⅓ 4
Explanation: Rewrite your formula in slope-intercept form!
You will need to subtract ⅔ x from both sides to do this and then divide both sides by c
Then, set the coefficient for x equal to the given slope (6) and solve for c. Since the coefficient is a fraction, write 6 as 6/1 and cross-multiply

15. A line in the xy-plane that passes through the coordinate points (3, -6) and (-7, -4) will never intersect a line that is represented by which of the following equations?
x+5y=6
x+y/2 = 7
y-2x=-9
2y-x = -8

16. A line in the xy-plane that passes through the coordinate points (3, -6) and (-7, -4) will never intersect a line that is represented by which of the following equations?
x+5y=6
x+y/2 = 7
y-2x=-9
2y-x = -8
Reasoning: Use m= y2-y1
X2-x1 to determine the slope of the line
given in the question.
You should come up with a slope of -⅕
Next, identify the slope in the answer choices
Remember: Parallel lines have = slope!

17. Let’s continue in your books!
Open to page 40 and complete questions 1-10 of the “extra practice problems.”
We will check them tomorrow!