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Identify Dependent Systems by Graphing Coincident Lines
Learn to spot dependent systems: graph two equations, see one line, confirm coincidence algebraically. Includes worked example, practice, rubric, and planned printable.
Free lesson
Four exercises
Worked answers
Algebra and Function Practice
A system of two linear equations is dependent when both equations describe the same line. On a graph, the two lines coincide: every point on one line is also on the other, so the system has infinitely many solutions. This lesson teaches you to recognize coincidence visually and confirm it algebraically. You will graph pairs of equations, compare slopes and intercepts, and check whether one equation is a multiple of the other. The focus is only on dependent systems; parallel inconsistent systems and substitution or elimination solving are outside this lesson. All numerical values used in examples are hypothetical and chosen for easy checking.
Practice materials: Printable downloads are not available for this lesson yet. Use the instructions and four exercises below.
What Coincident Lines Look Like
When you graph two linear equations on the same coordinate grid, coincident lines lie exactly on top of each other. You see only one line, but it represents both equations. Every point on that line satisfies both equations, so the solution set is the entire line: infinitely many solutions.
To confirm coincidence visually, plot at least three points for each equation using a simple table of values. If all plotted points from the second equation land on the same drawn line as the first, the lines are coincident. Because hand-drawn graphs can hide small differences, always follow the visual check with an algebraic check.
- Write each equation in slope-intercept form y = mx + b.
- Compare the slopes m. If the slopes differ, the lines intersect once and the system is independent, not dependent.
- If the slopes are equal, compare the y-intercepts b. Equal slopes and equal intercepts mean the lines coincide.
- Plot three points for each equation on the same grid and verify they lie on one line.
- Confirm algebraically that one equation is a nonzero multiple of the other.
Algebraic Confirmation of Dependence
Graphing gives a picture, but algebra gives certainty. Two equations in standard form Ax + By = C are dependent exactly when one equation can be multiplied by a nonzero constant to produce the other. For example, 2x + 3y = 6 and 4x + 6y = 12 are dependent because multiplying the first equation by 2 gives the second.
A quick arithmetic check: write both equations in the same form and compare the ratios of corresponding coefficients. If A1/A2 = B1/B2 = C1/C2, the lines coincide. Use simple whole numbers so the ratios are easy to compute by hand.
- Rewrite both equations in standard form Ax + By = C with integer coefficients if possible.
- Compute the ratio of the x-coefficients, the ratio of the y-coefficients, and the ratio of the constants.
- If all three ratios are equal, the system is dependent and the lines coincide.
- If the coefficient ratios are equal but the constant ratio differs, the lines are parallel and the system is inconsistent, not dependent.
- State the conclusion: the system has infinitely many solutions because the equations describe the same line.
Reading a Graph to Decide Dependence
A graph can show coincidence clearly when you use a consistent scale and label at least two points per line. Suppose a grid shows a line passing through (0, 1) and (2, 5). A second equation, y = 2x + 1, produces the same two points, so the lines coincide. The visible line is one line, but it satisfies both equations.
Be careful not to confuse coincidence with near-coincidence. If two lines have slopes that are close but not equal, they will cross somewhere off the visible grid. Always compute the slopes exactly from the equations rather than relying on the drawing alone.
- Choose a consistent scale and label the axes.
- Plot two or three points for each equation using a table of values.
- Draw each line and note whether the second line lies exactly on the first.
- Compute both slopes from the equations to confirm they are equal.
- Compute both y-intercepts to confirm they are equal.
Worked example
Graph the system y = 3x - 2 and 6x - 2y = 4. Determine whether the system is dependent by graphing and confirm algebraically.
Worked answer
The system is dependent; the lines coincide and there are infinitely many solutions.
Why: Rewrite 6x - 2y = 4 in slope-intercept form: subtract 6x to get -2y = -6x + 4, then divide by -2 to get y = 3x - 2. Both equations are y = 3x - 2, so they have the same slope 3 and the same y-intercept -2. Plotting points for each equation, such as (0, -2), (1, 1), and (2, 4), places all points on the same line. Algebraically, multiplying y = 3x - 2 by 2 gives 2y = 6x - 4, which rearranges to 6x - 2y = 4, so one equation is a nonzero multiple of the other.
Check: Substitute two different points from the line into both original equations. For (1, 1): 1 = 3(1) - 2 is true, and 6(1) - 2(1) = 4 is true. For (2, 4): 4 = 3(2) - 2 is true, and 6(2) - 2(4) = 4 is true. Both points satisfy both equations, confirming coincidence.
Four practice exercises
Exercise 1
Graph y = 2x + 1 and 4x - 2y = -2. Decide whether the system is dependent. Confirm algebraically.
Show answer and check
Dependent; the lines coincide and there are infinitely many solutions.
Why: Rewrite 4x - 2y = -2: subtract 4x to get -2y = -4x - 2, then divide by -2 to get y = 2x + 1. Both equations are y = 2x + 1, so slopes are both 2 and y-intercepts are both 1. Plotting points such as (0, 1), (1, 3), and (2, 5) for each equation gives the same line. Multiplying y = 2x + 1 by 2 gives 2y = 4x + 2, which rearranges to 4x - 2y = -2, confirming dependence.
Check: Substitute (1, 3) into both equations: 3 = 2(1) + 1 is true, and 4(1) - 2(3) = -2 is true. Substitute (2, 5): 5 = 2(2) + 1 is true, and 4(2) - 2(5) = -2 is true.
Exercise 2
Graph x + y = 3 and 2x + 2y = 6. Are the lines coincident? Show an algebraic check.
Show answer and check
Yes, the lines are coincident; the system is dependent with infinitely many solutions.
Why: Rewrite x + y = 3 as y = -x + 3. Rewrite 2x + 2y = 6 by dividing by 2 to get x + y = 3, then y = -x + 3. Both lines have slope -1 and y-intercept 3. Plotting (0, 3), (1, 2), and (2, 1) for each equation places all points on the same line. The second equation is exactly 2 times the first, so the equations are dependent.
Check: Check (1, 2): 1 + 2 = 3 is true, and 2(1) + 2(2) = 6 is true. Check (2, 1): 2 + 1 = 3 is true, and 2(2) + 2(1) = 6 is true.
Exercise 3
Graph y = -0.5x + 4 and x + 2y = 8. Determine whether the system is dependent and explain how the graph shows it.
Show answer and check
Dependent; the lines coincide and there are infinitely many solutions.
Why: Rewrite x + 2y = 8: subtract x to get 2y = -x + 8, then divide by 2 to get y = -0.5x + 4. Both equations have slope -0.5 and y-intercept 4. Plotting points such as (0, 4), (2, 3), and (4, 2) for each equation gives the same line. The graph shows one line because the two lines lie exactly on top of each other. Algebraically, multiplying y = -0.5x + 4 by 2 gives 2y = -x + 8, which rearranges to x + 2y = 8.
Check: Substitute (2, 3): 3 = -0.5(2) + 4 is true, and 2 + 2(3) = 8 is true. Substitute (4, 2): 2 = -0.5(4) + 4 is true, and 4 + 2(2) = 8 is true.
Exercise 4
Graph 3x - y = 2 and 6x - 2y = 4. A student says the lines are parallel because the coefficients are doubled. Are they correct? Confirm with slopes and intercepts.
Show answer and check
The student is incorrect; the lines coincide and the system is dependent with infinitely many solutions.
Why: Rewrite 3x - y = 2 as y = 3x - 2. Rewrite 6x - 2y = 4: subtract 6x to get -2y = -6x + 4, then divide by -2 to get y = 3x - 2. Both lines have slope 3 and y-intercept -2, so they coincide. Doubling all coefficients and the constant produces an equivalent equation, not a parallel distinct line. Plotting (0, -2), (1, 1), and (2, 4) for each equation confirms one line.
Check: Check (1, 1): 3(1) - 1 = 2 is true, and 6(1) - 2(1) = 4 is true. Check (2, 4): 3(2) - 4 = 2 is true, and 6(2) - 2(4) = 4 is true.
Common mistakes
- Assuming two lines that look close on a hand-drawn graph are coincident. Compute the slopes exactly from the equations. If the slopes differ at all, the lines intersect once and the system is not dependent.
- Confusing dependent systems with inconsistent parallel systems. Dependent lines have equal slopes and equal intercepts, so they coincide. Parallel inconsistent lines have equal slopes but different intercepts, so they never meet.
- Checking only one point to decide coincidence. One shared point only shows the lines intersect there. Check at least two points and compare slopes and intercepts algebraically.
- Multiplying only one side of an equation when testing for a multiple. Multiply the entire equation, both sides, by the same nonzero constant. For example, 2(x + y = 3) gives 2x + 2y = 6.
Self-check rubric
- Correctly rewrites both equations in slope-intercept form or standard form.
- Compares slopes and intercepts accurately and states whether they are equal.
- Plots at least two points per equation on a consistent grid and describes the coincidence.
- Confirms algebraically that one equation is a nonzero multiple of the other.
- States the conclusion: dependent system with infinitely many solutions.
- Uses clear labels and consistent units; arithmetic is checkable and correct.
Questions and answers
How many solutions does a dependent system have?
Infinitely many, because every point on the coincident line satisfies both equations.
Can two different-looking equations describe the same line?
Yes. For example, x + y = 4 and 2x + 2y = 8 describe the same line because the second is twice the first.
Do I always need to graph to identify a dependent system?
No. Comparing slopes and intercepts, or checking whether one equation is a multiple of the other, is sufficient. Graphing is a useful visual check.
What is the difference between dependent and inconsistent systems?
Dependent lines coincide and have infinitely many solutions. Inconsistent lines are parallel and distinct, so they have no solution.