UNIT 5Systems of equationsLesson 1 of 31 Solving Systems by Graphing2 Solving Systems by Substitution and Elimination3 Systems of Equations Word Problems
Solving Systems by Graphing
A system is two equations at once. Its solution is the point that is on both lines. On a graph, that is where the lines cross.
What you'll be able to do
- Check if a point solves both equations.
- Graph two lines and read where they cross.
- Tell from the slopes how many solutions there are.
- Say what the crossing means in a story.
The big idea
One point, two lines. A solution of a system makes both equations true. On a graph, it is on both lines, so it is where they cross.
Check both. Put the point into each equation. If either one is false, it is not a solution.
Graph each line. Plot (0, b) first. Then use the slope to plot the next point. Draw the line. Do the same for the second line.
An example. Graph y = x + 1 and y = −2x + 7. The lines cross at (2, 3). Check: 2 + 1 = 3, and −4 + 7 = 3.
No solution. Lines with the same slope and different y-intercepts are parallel. They never cross, so there is no solution.
Infinitely many. Two equations can be the same line written in different ways. Every point on it is a solution.
Between grid points. A crossing can sit between grid lines. Then a graph gives a good estimate. Check it in both equations.
Words you need
- System of equations
- Two equations that must both be true at once.
- Solution of a system
- A point that makes both equations true. It is where the lines cross.
- Parallel lines
- Lines with the same slope that never cross.
- Slope
- How steep a line is: the rise divided by the run.
- Y-intercept
- Where a line crosses the y-axis, at (0, b).
How many solutions?
| The lines | Slopes and y-intercepts | Solutions |
|---|---|---|
| Cross once | Different slopes | One |
| Parallel | Same slope, different y-intercepts | None |
| The same line | Same slope, same y-intercept | Infinitely many |
Worked examples
Example 1
Is (1, 4) a solution of y = 2x + 2 and y = −x + 5?
- First equation: 2(1) + 2 = 4. True.
- Second equation: −1 + 5 = 4. True.
- Both are true, so (1, 4) is a solution.
Answer: Yes, (1, 4) is a solution
Example 2
Graph y = (2/3)x − 1 and y = −x + 4. Where do they cross?
- Line 1: plot (0, −1). Go right 3 and up 2 to (3, 1).
- Line 2: plot (0, 4). Go right 1 and down 1 to (1, 3).
- The lines cross at (3, 1).
- Check: (2/3)(3) − 1 = 1, and −3 + 4 = 1.
Answer: (0, −1), (3, 1). (0, 4), (1, 3). They cross at (3, 1)
Example 3
How many solutions do y = 2x + 1 and y = 2x − 3 have?
- Both slopes are 2.
- The y-intercepts are 1 and −3. They are different.
- So the lines are parallel. There is no solution.
Answer: No solution
Common mistakes
Checking only one equation. A solution must make both equations true.
Writing the crossing as (y, x). Read across for x first, then up for y.
Calling lines with the same slope the same line. Check b too. Different b means parallel lines and no solution.
Taking a graph estimate as exact. A crossing between grid lines is an estimate. Check it in both equations.
Practice: 22 questions
12 of 22 are word problems. For a word problem, sketch it first and mark what you know.
Tier 1 Foundations Use the rule directly
- Adult tickets cost $12 and child tickets cost $8. A family buys 6 tickets for $56. So 12x + 8y = 56 and x + y = 6. Is (2, 4) a solution? Word problem
- Tomas says (2, 5) solves y = 3x − 1 and y = −x + 6, because 3(2) − 1 = 5. Find his mistake.
- Take y = x + 1. Give its y-intercept and the next whole-number point to the right. Do the same for y = −2x + 7. Graph both. Where do they cross?
- Ben has $10 and saves $5 a week. Ava has $30 and saves $1 a week. Graph both. When do they have the same amount? Word problem
- Jon starts at 0 m and climbs 200 m an hour. Kay starts at 600 m and goes down 100 m an hour. When are they at the same height? Word problem
Tier 2 Intermediate One more step to think through
- Ravi and Lena each read 30 pages a day. Ravi starts on page 40. Lena starts on page 15. Will Lena ever be on the same page as Ravi? Word problem
- How many solutions does the system 2x + y = 3 and y = −2x + 3 have?
- Two cyclists ride at 4 m each second. One starts 1 m past a line, the other 3 m behind it. Iris says y = 4x + 1 and y = 4x − 3 have infinitely many solutions. Find her mistake. Word problem
- Tank A has 1 L and gains 1/2 L a minute. Tank B has 4 L and loses 1 L a minute. Graph both. When do they hold the same amount? Word problem
- How many solutions does each system have? A: y = 2x + 3 and y = 2x − 3. B: y = 2x + 3 and y = −2x + 3. C: y = 2x + 3 and 4x − 2y = −6.
- Pool A has 2 ft of water and rises 1 ft every 3 hours. Pool B has 6 ft and drains 1 ft each hour. For each pool, give the first point and the next whole-number point you plot. When are the depths equal? Word problem
Tier 3 Advanced Several steps - find the hidden structure
- Shop A rents bikes for $12 plus $3 an hour. Shop B charges $6 an hour. Graph both. When is Shop A cheaper? Word problem
- Elevator A is at floor 10 and goes down 1 floor each second. Elevator B is at floor 3 and goes up 1 floor each second. Graph both. When are they level? Word problem
- The lines y = 2x + b and y = −x + 7 cross at (2, k). Find k and b.
- Sofia graphs y = 3x − 2 and y = −x + 5. She reads the crossing as (2, 4). Check her point.
- Taxi A charges $4 plus $2 a mile. Taxi B charges $1 plus $2.50 a mile. Where do they cost the same? Which is cheaper for 8 miles? Word problem
- Graph x + 2y = 8 and y = x − 2. Give two points you plot for each line. Where do they cross?
Tier 4 Challenge Stretch problems, some above grade level
- Start with y = −2x + 5. Write a second equation for each case. (a) The system has no solution. (b) It has infinitely many. (c) Its only solution is (1, 3).
- On a small grid, y = 0.25x + 3 and y = 0.24x + 1 look parallel. Jade says they never meet. Is she right?
- Plan A costs $30 plus $5 per GB. Plan B costs $50 plus $k per GB. The plans cost the same at 8 GB. Find k. Word problem
- Three hikers walk one trail. After x hours, Asha is at the x + 1 km mark. Ben is at the 5 − x km mark. Cal is at the 0.5x + 2 km mark. Are all three ever at the same mark? Word problem
- A point solves a system of inequalities when it makes both true. Graph y ≥ 2x − 1 and y < −x + 5. Which boundary line is solid, and which is dashed? Which of (1, 1), (2, 3) and (0, 3) solve both? Above grade level
Answers
Tier 1 Foundations
- Yes, (2, 4) is a solution. Check both: 12(2) + 8(4) = 24 + 32 = 56, and 2 + 4 = 6. Both are true.
- No, (2, 5) is not a solution. Tomas checked only one equation. In y = −x + 6, −2 + 6 = 4, not 5. So (2, 5) is not on both lines. (1.75, 4.25) makes both equations true: check it in y = 3x − 1 and y = −x + 6.
- (0, 1), (1, 2). (0, 7), (1, 5). They cross at (2, 3). y = x + 1: start at the y-intercept (0, 1). The slope is 1, so go up 1 and right 1 each time: (1, 2). y = −2x + 7: start at the y-intercept (0, 7). The slope is −2, so go down 2 and right 1 each time: (1, 5). Where the lines meet, (2, 3) is on both. Check it in both equations.
- After 5 weeks, when both have $35. (5, 35) makes both equations true: check it in y = 10 + 5x and y = 30 + x.
- After 2 hours, at 400 m. (2, 400) makes both equations true: check it in y = 200x and y = 600 − 100x.
Tier 2 Intermediate
- No, the lines are parallel. Both lines have slope 30. Their y-intercepts are 40 and 15. Parallel lines never cross.
- Infinitely many solutions. Get y alone in 2x + y = 3: y = −2x + 3. That is the second equation. They are the same line.
- No solution. Iris looked only at the slopes. The y-intercepts are 1 and −3, so the lines are parallel. There is no solution. The lines y = 4x + 1 and y = 4x − 3 are parallel (same slope, different intercepts), so they never meet.
- After 2 minutes, when both hold 2 L. (2, 2) makes both equations true: check it in y = 1 + x/2 and y = 4 − x.
- A has none. B has one, (0, 3). C has infinitely many.. A: same slope, different y-intercepts, so parallel. B: different slopes, so they cross once, at (0, 3). C: 4x − 2y = −6 gives y = 2x + 3, the same line.
- (0, 2), (3, 3). (0, 6), (1, 5). They cross at (3, 3). Pool A starts at (0, 2). Its slope is 1/3, so go right 3 and up 1 to (3, 3). Pool B starts at (0, 6) and goes right 1, down 1 to (1, 5). They meet at 3 hours, at 3 ft. Check it in both equations.
Tier 3 Advanced
- For more than 4 hours. The lines cross at 4 hours, $24. The lines cross at (4, 24). After that, Shop B's steeper line is higher. So Shop A is cheaper for more than 4 hours.
- At 3.5 seconds, halfway between floors 6 and 7. The lines cross at (3.5, 6.5), a half-grid point. Check: 10 − 3.5 = 6.5, and 3 + 3.5 = 6.5.
- k = 5 and b = 1. k comes from y = −x + 7: k = −2 + 7 = 5. Then 5 = 2(2) + b, so b = 1.
- No, (2, 4) is not a solution. (2, 4) works in y = 3x − 2 but not in y = −x + 5, since −2 + 5 = 3. The lines cross between grid points, near (1.75, 3.25). (1.75, 3.25) makes both equations true: check it in y = 3x − 2 and y = −x + 5.
- At 6 miles, for $16. Taxi A is cheaper for 8 miles. The lines cross at (6, 16). For 8 miles, Taxi A costs 4 + 16 = $20 and Taxi B costs 1 + 20 = $21.
- (0, 4), (8, 0). (0, −2), (1, −1). They cross at (4, 2). For x + 2y = 8, put x = 0 to get (0, 4) and y = 0 to get (8, 0). For y = x − 2, plot (0, −2) and (1, −1). The lines cross at (4, 2). Check it in both equations.
Tier 4 Challenge
- For example: (a) y = −2x + 1. (b) 2y = −4x + 10. (c) y = x + 2.. Same slope and new b gives parallel lines. Doubling both sides gives the same line. A new slope through (1, 3) crosses only there.
- No. The slopes differ, so the lines cross. The slopes are 0.25 and 0.24. They are close but not equal, so the lines must cross. They meet far off the grid, at (−200, −47). Different slopes always cross once. Here the slopes differ by only 0.01, so the crossing is far away.
- k = 2.5. The crossing is at 8 GB. Plan A costs 30 + 5(8) = $70 there. Plan B must too: 50 + 8k = 70, so 8k = 20 and k = 2.5.
- Yes: after 2 hours, all three are at the 3 km mark. Asha and Ben meet when x + 1 = 5 − x, at 2 hours and the 3 km mark. Cal is at 0.5(2) + 2 = 3 km then too. So all three meet.
- y = 2x − 1 is solid. y = −x + 5 is dashed. Solutions: (1, 1) and (0, 3). (1, 1) is on the solid line y = 2x − 1, so it counts. (2, 3) is on the dashed line y = −x + 5, so it doesn't.
Stuck on this topic?
Trouble here often traces back to an earlier skill. Our free check-in finds the exact gap. A tutor then works on it with your child.
Find gaps free More practice