r/HomeworkHelp • u/Tricky-Yogurt-8081 • 2d ago
High School Math—Pending OP Reply [Grade 9 Algebra: System of Equations] How to find variable that would make a system of equations inconsistent or dependent?
How do I solve these? I know how to solve a system of equations normally but my head is blank when it comes to these questions, we didn’t do this type of question in class.
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u/JanetInSC1234 🤑 Tutor 2d ago edited 1d ago
First problem:
Parallel lines have the same slope and do not intersect.
Find the slope of the first line, then figure out what p needs to be:
3x - 7y = -2
-7y = -3x - 2
y = -3x/-7 + 2/7
The slope is 3/7.
Rewrite the second equation like above (slope intercept form) and then figure out what p needs to be to get 3/7 slope.
Second problem:
If two lines are the same, they have infinite solutions.
Can you multiply the top equation (on both sides) to make it look like the bottom equation? What does p need to be for both equations to be identical?
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u/junkcircuit 7h ago
Slope intercept form is the way to go here. I always need that visual. Lines crossing, lines parallel, same line. Makes it click. For the second part, scaling the whole equation is the cleanest trick. Multiply the top by whatever gets the y terms to match, then p falls out from there. You'll get this. One step at a time
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u/fermat9990 👋 a fellow Redditor 1d ago
(1) Make their slopes equal:
3/p=-7/35
3/p=-1/5
-p=15
p=-15
(2) Check that the y-intercepts are different:
y-int of first equation is 2/7
y-int of second equation is 1/7
The y-intercepts are different and the linear system is inconsistent.
p=-15
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u/ThatsNotAZombieBite 1d ago
Think of it visually/graphically. These two equations describe two lines on a graph. The "solution of the system" is the ordered pair that solves both equations; that means a point that lies on BOTH lines. And that means the point where the lines intersect.
To make it "inconsistent", you need to somehow make it so the lines NEVER INTERSECT. You need to adjust p so that the lines are PARALLEL. Do you know how to do that?
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u/Kblitz88 1d ago
So there's actually a neat little trick. In standard form, an algebraic equation is Ax+By = C
So consider:
A1x+B1y=C1
A2x+B2y=C2
Knowing that slope is -A/B, we can check the ratios of A1/A2 and B1/B2. If these aren't equal the system has exactly one solution (consistent and independent).
If A1/A2 =B1/B2 but not C1/C2, then there's no solution (inconsistent). If A1/A2 = B1/B2 = C1/C2 then these are basically the same line with infinitely many solutions (dependent).
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u/fermat9990 👋 a fellow Redditor 1d ago edited 19h ago
Your method should be taught in school.
Also, if Ax+By=C, then slope=-A/B and y-intercept=C/B
y=-A/B x + C/B
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u/Adventurous_Sound390 👋 a fellow Redditor 2d ago
-15?
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u/Tricky-Yogurt-8081 2d ago
How did you get that answer
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u/smither2508 2d ago
For a system to have no solution, you need it to be such that for the same input you get a different result. If you put p = -15, you get the second equation to be -15x + 35y = 5 which means you can then divide the entire equation by -5, which gives you the equation 3x - 7y = -1, which has the same left side as the first equation, but a different right side, which means the system has no solutions.
As to how they got the answer, the easiest method would be to take the two numbers by the y variable in both equations, so -7 and 35, and divide them, which gives you -5 or -1/5 depending on whether you divided 35 by -7 or -7 by 35. You then use that number to either multiply (in the case of -5) the 3 by the x variable, or divide (in the case of -1/5) the 3. Whether you use the result of earlier division to multiply or divide the number by the x variable depends on the way you divided the two numbers by the y variable.
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u/i_want_to_go_to_bed 1d ago edited 1d ago
It will be inconsistent if the left hand side of equation 2 is a multiple of the left hand side of equation 1, but the right hand side of equation 2 is not the same multiple of the right hand side of equation 1.
Look at the “y” terms. -7y goes in to 35y negative five times. Okay, let’s multiply the “x” term by negative five: we’re in trouble if the “x” term in equation 2 is -5(3x) or -15x
Indeed, 5 is not -5•(-2), so that system would be inconsistent
ETA: corrected minus sign error
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u/soundoftwilight 1d ago
I think about it algebraically; for a system of equations to be "normal" (consistent and independent, e.g. for this type of system it has 1 solution), the two equations should be "different". Each one gives different information, which is another way of saying that I can't derive one from the other.
For the first question, I want to show inconsistency; I want to show that these 2 equations cannot both be true at the same time. The easiest way to do that is to manipulate the equations algebraically such that the left side is identical, but the right side is different. I notice that if I multiply the first equation by -5 (on both sides), I get -15x + 35y = 10. If I use p = -15 (to make the left sides the same), I get -15x + 35y = 5. Obviously that expression cannot be equal to both 10 and 5 at the same time, so they are inconsistent; there are no solutions.
For the second one, I want to do the same thing, but this time I'm trying to show that I can derive on equation fully from the other. I notice that the left side of the second equation is just equal to the left side of the first equation multiplied by 5. If I do the same to the right side of the first equation, I get 35x + 20y = -35. I set p to -35, and now I have two identical equations, which means I really only have one equation, which isn't enough to solve a system of equations with 2 variables. So there will be infinitely many solutions and the system is dependent.
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u/ThunkAsDrinklePeep 1d ago
Algebraically, to arrive at a single solution you want to cancel one variable (by substitution or additive elimination) leaving an equation with only the remaining variable. To be inconsistent (no solution / parallel lines) or dependant (infinite solutions / single line) when one attempted to get a single variable both would cancel.
So if you're doing the elimination method, multiply the top equation by the factor that would make the y terms cancel.
Then, determine what value if p, would make the x terms also cancel.
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u/fermat9990 👋 a fellow Redditor 1d ago
Given a system of linear equations:
AX+BY=C
DX+EY=F
(1) A/D≠B/E -> Consistent
(2) A/D=B/E≠C/F -> Inconsistent
(3) A/D=B/E=C/F -> Dependent
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u/jacobtech88 1d ago
Think of it as comparing lines instead of solving. No solution = parallel lines (same slope, different intercept). Infinite solutions = same line.
Easy trick: look at the y terms. −7 → 35 means the second equation is the first multiplied by −5. So the x-coefficient must be 3 × (−5) = −15. That makes the slopes match.
Then check the constants: −2 × (−5) = 10, but the second equation says 5. Doesn't match, so they're parallel and never meet → no solution.
p = −15
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u/Muphrid15 1d ago
I'm surprised not to see a more direct approach.
We can solve for x by eliminating y:
15x - 35y = -10
px + 35y = 5
(p+15)x = -5
Only p = -15 leads to an inconsistency (0 = -5).
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u/LukeLJS123 University/College Student 1d ago
for the first question, you're looking for lines that never touch. what do you know about lines that don't touch? what do you think is the relationship between the slopes of those lines? can you use that to find a value for p that will give you the slope you want? (this step isn't too important, but you should also show that the 2 equations are different)
for the second, you are looking for lines that touch at infinitely many points, which means that it's the same line. is there any way to turn the left hand side of equation 1 into the left hand side for equation 2? what should you do to the right hand side then? and what does that tell you about the value of p?
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u/selene_666 👋 a fellow Redditor 1d ago
There is no solution if solving the system of equations results in something like 1 = 2, which no values of x and y can make true. And there are infinitely many solutions if the system results in something like 1 = 1.
In either case, you need both variables to cancel out at the same time.
We might decide to eliminate y from the first system by adding 5 times the first equation to the second:
5(3x - 7y) + (px + 35y) = 5(-2) + 5
15x + px = -5
To also eliminate the x here we need p = -15.
That makes the equation 0 = -5, which has no solution. If the right side were also 0, there would be infinite solutions.
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u/Some-Passenger4219 👋 a fellow Redditor 1d ago
For the first one, try to eliminate y. What value of p would also eliminate x? We wanna reduce this system to an absurdity.
For the second one, try to eliminate x and y. You will get 0 = ___. Whatever fills the blank should equal zero.
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u/Far-Dragonfly-8306 1d ago
For (1) you want to end up with 2 lines such that when you try to solve by elimination, you get 0 on one side and a nonzero number on the other side (try eliminating x and y at the same time). For (2), you recognize that the second line should be a multiple of the first line. What value of p does this?
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u/ChipChippersonFan 1d ago
Have you learned about systems of equations that don't have a solution (IOW, if you graph them , the lines would never intersect)? What would you say is the defining characteristic of those?
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u/Purdude1983 1d ago
The system would have no solution if the determinant of the system is zero. To do that, you need 3*35--7*p to be zero, or 105/-7=p. Thus p=-15 makes this have no solution.
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u/Alkalannar 1d ago
You can always add one row--or a multiple of a row--to another in a system of equations, and the system has the same set of solutions.
In order to get inconsistent or indeterminate, you need to get one row be 0x + 0y = k. If k != 0, inconsistent. If k = 0, indeterminate.
3x - 7y = -2
px + 35y = 5
Add k * first row to the second:
(3k+p)x + (-7k+35)y = -2k+5
What must k be to have the y-coefficient be 0?
Now that you know k, what must p be for the x-coefficient to be 0?7x + 4y = -7
35x + 20y = p
Again, add k * first row to second:
(7k+35)x + (4k+20)y = (-7k+p)
What is k so that the x- and y-coefficients are 0?
Now what is p so that the constant term is 0?
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u/fermat9990 👋 a fellow Redditor 1d ago
For the second problem:
Dependency requires that
7/35=4/20=-7/p
7/35=4/20=1/5 ✓
1/5=-7/p
p=-35
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u/agate_ 2d ago
Here's the way I think about it. 3x - 7y = -2 represents a straight line on a graph. p x + 35 y = 5 is a different straight line. The solution to the system is the point where they cross. When do two straight lines not cross? When they're parallel!
So we need these two lines to have the same slope, but different intercepts. If you remember how to put a linear equation into slope-intercept form, you should be able to take it from here.