NOTESNOTESParallel Lines and Parallel Lines and

Word ProblemsWord Problems

Parallel LinesParallel Lines

Lines that Lines that never never intersectintersect We can tell two linesWe can tell two lines are parallel if we graph are parallel if we graph them them OROR If we look at their If we look at their slopeslope What do we notice aboutWhat do we notice about the slope of the lines on the the slope of the lines on the left?left?

y

5

5

-5

-5

Slopes are the Slopes are the same!same!

Example #1Example #1

Decide whether the graphs of the two Decide whether the graphs of the two equations are parallel lines. Explain equations are parallel lines. Explain your reasoning.your reasoning.

line aline a: -: -xx + 2 + 2yy = 6 = 6 line b:line b: - -xx + 2 + 2yy = -2 = -2

Which means?Which means?

Look to see if they have same slope.Look to see if they have same slope.

Step 1: rewrite equations in Step 1: rewrite equations in y = mx + by = mx + b

Step 2: compare the slopesStep 2: compare the slopes

TRY ON YOUR OWNTRY ON YOUR OWN

Example #1 cont.Example #1 cont.Decide whether the graphs of the two equations are Decide whether the graphs of the two equations are

parallel lines. Explain your reasoning.parallel lines. Explain your reasoning. line aline a: -: -xx + 2 + 2yy = 6 = 6 line b:line b: - -xx + 2 + 2yy = -2 = -2

+x +x+x +x +x +x+x +x

2y2y = = xx++66 2y2y = = xx - - 22 2 2 2 2 2 2 2 2

2 22 2

Line aLine a: : y = ½xy = ½x + 3 + 3 Line bLine b: : y = ½xy = ½x – 1 – 1

YesYes, lines are parallel since they have the same slope., lines are parallel since they have the same slope.

Example #2Example #2

Decide whether the graphs of the Decide whether the graphs of the two equations are parallel lines. two equations are parallel lines. Explain your reasoning.Explain your reasoning.

line aline a: 3y = -9x – 5 : 3y = -9x – 5 line b:line b: 2y – 2y – 6x = -56x = -5

TRY ON YOUR OWNTRY ON YOUR OWN

Example #2 cont.Example #2 cont.

Decide whether the graphs of the two Decide whether the graphs of the two equations are parallel lines. Explain your equations are parallel lines. Explain your reasoning.reasoning.

line aline a: 3y = -9x – 5 : 3y = -9x – 5 line b:line b: 2y – 6x = -5 2y – 6x = -5

line aline a: m = -3 : m = -3 line b:line b: m = 3 m = 3

NONO, lines are not parallel. The slope of the , lines are not parallel. The slope of the lines are different.lines are different.

53

3y x

53

2y x

Example #3Example #3

A submarine started to A submarine started to decrease from a depth of decrease from a depth of 5 feet below the water at 5 feet below the water at a rate of ½ feet per a rate of ½ feet per hour. The equation y = –hour. The equation y = –½ ½ x – 5x – 5 models the depth models the depth y of the submarine after y of the submarine after xx hours. hours.

Example #3Example #3

A submarine started to decrease A submarine started to decrease from a depth of 5 feet below the from a depth of 5 feet below the water at a rate of ½ feet per hour. water at a rate of ½ feet per hour. The equation y = – ½ The equation y = – ½ x – 5x – 5 models models the depth y of the submarine after the depth y of the submarine after xx hours. hours.

a)a) What is the slope of y = –½ What is the slope of y = –½ x – 5x – 5 ? ? What is the y-intercept?What is the y-intercept?

Slope = m = -½ Slope = m = -½

y-intercept = - 5y-intercept = - 5

Example #3Example #3

A submarine started to decrease from A submarine started to decrease from a depth of 5 feet below the water at a a depth of 5 feet below the water at a rate of ½ feet per hour. The equation rate of ½ feet per hour. The equation y = -½ y = -½ x – 5x – 5 models the depth y of the models the depth y of the submarine after submarine after xx hours. hours.

b)b) Explain what the slope and y-intercept Explain what the slope and y-intercept mean in relation to the problem.mean in relation to the problem.

Slope is the rate at which the submarine Slope is the rate at which the submarine is going down.is going down.

The y-intercept is the depth of the The y-intercept is the depth of the submarine at the very beginningsubmarine at the very beginning

Example Example #3#3C)C)Graph the depth Graph the depth

of the of the Submarine Submarine

over a 6 hour over a 6 hour period of period of timetime

Example #3Example #3

A submarine started to decrease from A submarine started to decrease from a depth of 5 feet below the water at a a depth of 5 feet below the water at a rate of ½ feet per hour. The equation rate of ½ feet per hour. The equation y = –½ y = –½ x – 5x – 5 models the depth y of models the depth y of the submarine after the submarine after xx hours. hours.

d)d) If the submarine gets a depth of 12 feet, If the submarine gets a depth of 12 feet, it will need repairing. Is it going to get it will need repairing. Is it going to get down to 12 feet in 6 hours?down to 12 feet in 6 hours?

Look at your graph…. No, after 6 hours the Look at your graph…. No, after 6 hours the submarine will only be at a depth of 8 submarine will only be at a depth of 8 feet below the waterfeet below the water