Depth · one concept from first principles · step 2 of 4
A second solution forces the two lines to coincide
The system of equations , has a solution other than , . Find the value of the constant .
Expressions
Your answer
How to solve
Strategy Convert to Algebra — First reorganize the second equation so both have the form (…)x + (…)y = 0. Picture them as lines: both pass through the origin. Two different lines can share only one point, so a second shared point means the lines are identical, which turns into a proportion of coefficients.
1STEP 1
Move everything to the left side
Rewrite the second equation so both equations have 0 on the right.
2x + y = kx → (2-k)x + y = 0
Now both equations have the same shape, which makes them easy to compare.
8.EE.C.8Organize Information In More Ways2STEP 2
Picture the two lines
Both lines pass through the origin. Two different lines meet at most once, so another common solution means they are the same line.
One shared point is automatic here; a second one forces everything to be shared.
8.EE.C.8Draw A DiagramA common solution other than can exist only when the two equations describe the very same line.
Why?
Both lines already pass through , because putting for and makes each side of each equation equal to .
🧱A solution makes the equation trueA solution is a number that, put in for the unknown, makes both sides come out equal; that is all being a solution means.
Why?
Two different lines cross at most once, so if these two shared a second point besides they would have to be the very same line.
🧱Two different lines cross at most onceTwo points fix a straight line, so two lines that shared two points would be the same line; different lines cross once or, if parallel, never.
3STEP 3
Make the coefficients proportional
Two equations of the form (…)x+(…)y=0 describe the same line when their x-coefficients and y-coefficients are in the same ratio.
k/(2-k) = -3/1
Same line means one equation is a constant multiple of the other.
8.EE.C.8Convert To Algebra4STEP 4
Solve for k
Cross-multiply and collect the k terms.
k = -3(2-k) = -6 + 3k → -2k = -6 → k = 3
A one-variable linear equation finishes the job.
8.EE.C.8Convert To AlgebraAnswer
3
With k=3 the equations are 3x-3y=0 and 2x+y=3x, and both simplify to y=x. So x=1, y=1 (and every point with y=x) is a solution besides the origin.
Takeaway
Two lines through the same point share another point only if they are the same line.
- Move everything to the left side
- Picture the two lines
- Make the coefficients proportional
- Solve for k
Where next?
Level upsuggested
Another one like thissuggested
Step back and solidifysuggested
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