Grade by Grade
Analytic Geometry · Analytic Geometry

Choosing a Method and Predicting the Number of Solutions

1.1 1.2About 65 minutes
Learning goals
  • I can choose an efficient method (graphing, substitution or elimination) for a linear system and explain why.
  • I can predict whether a system has one solution, no solution or infinitely many solutions by comparing slopes and y-intercepts.
  • I can find a parameter value that makes a system have no solution or infinitely many solutions.

Words to know

efficient
Getting a correct answer with the fewest steps and least chance of error.
slope-intercept form
y = mx + b, where m is the slope and b is the y-intercept.
parameter
A letter such as k that stands for a fixed number that you may choose or find.
inconsistent system
A system with no solution (parallel lines).
consistent system
A system with at least one solution.

Three tools, one job

You now have three ways to solve a linear system: graphing, substitution and elimination. A carpenter owns a hammer, a saw and a drill, and chooses the right one for each job. A good mathematician does the same. In this lesson you will learn how to pick an efficient method, and how to predict the number of solutions before you start.

MethodBest when...Watch out for...
Graphingyou want to see the situation, compare plans, or estimate; technology is availableinexact answers when the crossing point is not on a grid point
Substitutionone equation is already solved for a variable, or a variable has a coefficient of 1 or −1fractions if you isolate a variable with another coefficient; missing brackets
Eliminationboth equations are in standard form ax + by = cmultiplying every term; sign errors when subtracting
Choosing a method.
Is a variable already isolated,or has a coefficient of 1 or −1?yesSubstitutionnoBoth equations in the formax + by = c?yesEliminationneed a picture or estimate?Graphing (or technology)Every method gives the same answer.Always check in both equations.
A quick decision guide. It is a guide, not a rule: any correct method earns full marks, but an efficient one saves time and avoids errors.
Worked example

Example 1: Choose a method

Choose an efficient method for each system and explain your choice. (a) y = 4x − 1 and 3x + 2y = 20. (b) 5x + 3y = 11 and 2x − 3y = 17. (c) Two cell phone plans you want to compare on a graph.

  1. (a) Substitution: y is already isolated. 3x + 2(4x − 1) = 20, 11x = 22, x = 2, y = 7. Check: 6 + 14 = 20 ✓.
  2. (b) Elimination: both are in standard form and +3y and −3y are opposites. Add: 7x = 28, x = 4. Then 20 + 3y = 11, y = −3. Check: 8 + 9 = 17 ✓.
  3. (c) Graphing: a graph shows which plan is cheaper on each side of the crossing point, not just where they are equal.

(a) substitution, (2, 7); (b) elimination, (4, −3); (c) graphing.

Predicting the number of solutions

Rewrite each equation in slope-intercept form, y = mx + b, and compare the slopes and the y-intercepts.

  • Different slopes → the lines cross once → exactly one solution.
  • Same slope, different y-intercepts → parallel → no solution.
  • Same slope, same y-intercept → the same line → infinitely many solutions.

Shortcut for standard form. If one equation is a multiple of the other on the left side only, the lines are parallel. If the whole equation (including the right side) is a multiple, they are the same line. For example, 2x − 3y = 6 and 4x − 6y = 12 are the same line (multiply the first by 2), but 2x − 3y = 6 and 4x − 6y = 5 are parallel.

Worked example

Example 2: How many solutions?

Without solving, state the number of solutions. (a) 2x − 3y = 6 and 4x − 6y = 12. (b) 2x − 3y = 6 and 4x − 6y = 5. (c) x + y = 5 and x − y = 1.

  1. (a) Rearrange: −3y = −2x + 6, so y = (2/3)x − 2. And −6y = −4x + 12, so y = (2/3)x − 2. Same slope, same intercept: the same line.
  2. (b) The first is y = (2/3)x − 2. The second: −6y = −4x + 5, so y = (2/3)x − 5/6. Same slope, different intercepts: parallel.
  3. (c) y = −x + 5 has slope −1 and y = x − 1 has slope 1. Different slopes: they cross once.

(a) infinitely many; (b) none; (c) exactly one (it is (3, 2)).

Worked example

Example 3: Find a value that makes a system special

For what value of k does the system kx + 2y = 8 and 3x + y = 5 have no solution?

  1. Write both in slope-intercept form. First: 2y = −kx + 8, so y = −(k/2)x + 4. Second: y = −3x + 5.
  2. For no solution the slopes must be equal: −k/2 = −3, so k = 6.
  3. Check the intercepts: they are 4 and 5, which are different, so the lines really are parallel (not the same line).
  4. Check with elimination: 6x + 2y = 8 and (×2) 6x + 2y = 10. Subtract: 0 = −2, false ✓.

k = 6. For any other value of k the slopes differ, so there is exactly one solution.

−3−2−11234−3−2−1123450xy(0, 4)(0, 5)6x + 2y = 83x + y = 5slope −3for both
With k = 6, the lines 6x + 2y = 8 (solid, y = −3x + 4) and 3x + y = 5 (dashed, y = −3x + 5) have the same slope and different y-intercepts, so they never meet.
Worked example

Example 4: Same answer, different routes

Solve 4x − 3y = 1 and y = 2x − 3 by substitution, then by elimination, and compare.

  1. Substitution: 4x − 3(2x − 3) = 1, so 4x − 6x + 9 = 1, −2x = −8, x = 4, y = 2(4) − 3 = 5.
  2. Elimination: rewrite the second as 2x − y = 3, then multiply by 3: 6x − 3y = 9. Subtract 4x − 3y = 1: 2x = 8, x = 4, then y = 5.
  3. Check: 4(4) − 3(5) = 16 − 15 = 1 ✓ and 2(4) − 3 = 5 ✓.
  4. Substitution took fewer steps here because y was already isolated.

Both methods give (4, 5). Substitution was more efficient for this system.

Just for fun

What did one line say to the line lying exactly on top of it?

"We have infinitely much in common."

Hands-on activity

Method race

You need: paper, a pencil, a timer (a phone or kitchen timer) and graph paper.

  1. Copy these systems: (A) y = x + 2 and 3x + y = 14. (B) 2x + 3y = 12 and 4x − 3y = 6. (C) y = 2x and y = −x + 6.
  2. Before solving, predict which method will be fastest for each and write your prediction.
  3. Time yourself solving each system with your chosen method. Then solve one of them with a different method and time that too.
  4. Check each answer in both equations.
  5. Write two sentences: which method was fastest for which kind of system, and did your predictions hold?

Answers to check against: (A) (3, 5); (B) (3, 2); (C) (2, 4). Substitution usually wins for A and C; elimination usually wins for B.

Your turn

1

Choose a method and solve: y = 3x + 1 and 2x + y = 11.

2

Choose a method and solve: 3x + 4y = 10 and 3x − 2y = 4.

3

How many solutions: y = −2x + 3 and y = −2x − 1?

4

How many solutions: 3x + y = 7 and 9x + 3y = 21?

5

How many solutions: y = 5x + 2 and y = −5x + 2?

6

For what value of m are y = mx + 4 and 6x − 2y = 3 parallel?

More practice: warm-up

1

Rewrite 2x + y = 9 in the form y = mx + b.

2

What is the slope of 3x − y = 5?

3

Lines with equal slopes and different y-intercepts are called...

4

Which method would you use for x = 5y and 2x + 3y = 26?

5

Which method would you use for 7x + 2y = 3 and 5x − 2y = 9?

More practice: on your own

1

State the number of solutions: x − 2y = 4 and −3x + 6y = −12.

2

State the number of solutions: 4x + 6y = 9 and 6x + 9y = 9.

3

Solve by any method: x/2 + y = 4 and 3x − 2y = 8.

4

Find k so that 2x + ky = 6 and x + 3y = 3 have infinitely many solutions.

5

Solve by any method: 6x − 5y = 4 and 4x − 3y = 2.

Challenge

1

Find values of a and b so that ax + 3y = b and 4x − 6y = 10 have (i) infinitely many solutions and (ii) no solution.

2

A student says, "If two lines have different y-intercepts, the system has no solution." Give a counterexample and correct the statement.

3

Solve: 3(x − 1) + 2(y + 2) = 15 and (x + y)/2 − (x − y)/4 = 2.

Check your understanding

Quiz

1.

Which method is most efficient for y = 2x − 7 and 5x − 3y = 19?

2.

Two lines have slopes −4 and −4 and y-intercepts 1 and −3. How many solutions does the system have?

3.

How many solutions does 5x − 2y = 3 and 10x − 4y = 6 have?

4.

For what value of k are kx + y = 2 and 4x + 2y = 9 parallel?

5.

Which statement is true?

6.

Which system has exactly one solution?

For parents and teachers

Big idea: all three methods find the same solution; good problem solvers pick the most efficient method and can predict whether a system has one, none or infinitely many solutions by comparing slopes and intercepts. This lesson supports expectations 1.1 (algebraic methods) and 1.2 (choosing an appropriate algebraic or graphical method).

How to help: For each system, ask "Which method would you pick, and why?" before your teen starts. The method race activity makes this concrete.

Common difficulties:

  • Thinking that different intercepts always means no solution (the slopes must also be equal).
  • Errors when rearranging into y = mx + b, especially dividing every term by a negative coefficient.
  • Using a long method out of habit. Both substitution and elimination are correct; the goal is efficiency and accuracy.