Understand System Solutions as Intersections - 8th Grade Math
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What is the solution of the system $y=x-5$ and $y=3x-9$?
What is the solution of the system $y=x-5$ and $y=3x-9$?
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$(2,-3)$. Set equal: $x-5=3x-9$, giving $x=2$, then $y=-3$.
$(2,-3)$. Set equal: $x-5=3x-9$, giving $x=2$, then $y=-3$.
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Identify whether $(1,2)$ is the intersection of $y=2x$ and $y=x+1$.
Identify whether $(1,2)$ is the intersection of $y=2x$ and $y=x+1$.
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Yes, $(1,2)$ satisfies both equations. Check: $2=2(1)$ ✓ and $2=1+1$ ✓.
Yes, $(1,2)$ satisfies both equations. Check: $2=2(1)$ ✓ and $2=1+1$ ✓.
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What is the number of solutions to the system $y=3x-1$ and $y=3x+2$?
What is the number of solutions to the system $y=3x-1$ and $y=3x+2$?
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No solution. Same slope, different intercepts = parallel.
No solution. Same slope, different intercepts = parallel.
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What is the number of solutions to the system $y=-2x+7$ and $y=-2x+7$?
What is the number of solutions to the system $y=-2x+7$ and $y=-2x+7$?
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Infinitely many solutions. Identical equations = same line.
Infinitely many solutions. Identical equations = same line.
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Identify whether $(2,5)$ is a solution to $y=2x+1$ and $y=-x+7$.
Identify whether $(2,5)$ is a solution to $y=2x+1$ and $y=-x+7$.
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Yes, $(2,5)$ satisfies both equations. Check: $5=2(2)+1$ ✓ and $5=-2+7$ ✓.
Yes, $(2,5)$ satisfies both equations. Check: $5=2(2)+1$ ✓ and $5=-2+7$ ✓.
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What is the intersection point of $x=4$ and $y=-3$?
What is the intersection point of $x=4$ and $y=-3$?
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$(4,-3)$. Vertical and horizontal lines meet at one point.
$(4,-3)$. Vertical and horizontal lines meet at one point.
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Which graph feature shows the solution to a linear system: the $y$-intercept or the intersection point?
Which graph feature shows the solution to a linear system: the $y$-intercept or the intersection point?
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The intersection point. Solutions appear where graphs meet, not at axes.
The intersection point. Solutions appear where graphs meet, not at axes.
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What does it mean if two lines lie on top of each other (coincide) on a coordinate plane?
What does it mean if two lines lie on top of each other (coincide) on a coordinate plane?
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The system has infinitely many solutions. Same line means every point satisfies both equations.
The system has infinitely many solutions. Same line means every point satisfies both equations.
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What does it mean if two lines intersect at exactly one point on a coordinate plane?
What does it mean if two lines intersect at exactly one point on a coordinate plane?
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The system has exactly one solution. Lines that cross once have one common point.
The system has exactly one solution. Lines that cross once have one common point.
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What does a solution $(x,y)$ to a system of two linear equations represent on the graphs?
What does a solution $(x,y)$ to a system of two linear equations represent on the graphs?
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The intersection point of the two lines at $(x,y)$. Where both lines meet, both equations are satisfied.
The intersection point of the two lines at $(x,y)$. Where both lines meet, both equations are satisfied.
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What does it mean if two lines are parallel and never intersect on a coordinate plane?
What does it mean if two lines are parallel and never intersect on a coordinate plane?
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The system has no solution. Parallel lines never meet, so no common points.
The system has no solution. Parallel lines never meet, so no common points.
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What must be true about a point $(x,y)$ for it to be the intersection of two lines in a system?
What must be true about a point $(x,y)$ for it to be the intersection of two lines in a system?
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It satisfies both equations when substituted. Intersection points make both equations true.
It satisfies both equations when substituted. Intersection points make both equations true.
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Identify the number of solutions if the graphs intersect at $(3,-2)$.
Identify the number of solutions if the graphs intersect at $(3,-2)$.
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One solution. Lines crossing at one point means one solution.
One solution. Lines crossing at one point means one solution.
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Identify the solution set if the two equations graph as the same line.
Identify the solution set if the two equations graph as the same line.
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Infinitely many solutions (all points on the line). Same line means every point is a solution.
Infinitely many solutions (all points on the line). Same line means every point is a solution.
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Identify the number of solutions if the graphs are parallel distinct lines.
Identify the number of solutions if the graphs are parallel distinct lines.
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No solution. Parallel lines never intersect.
No solution. Parallel lines never intersect.
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Which statement matches a system with one solution: the lines have the same slope or different slopes?
Which statement matches a system with one solution: the lines have the same slope or different slopes?
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Different slopes. Lines with different slopes must intersect once.
Different slopes. Lines with different slopes must intersect once.
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Which statement matches a system with no solution: same slope with different $y$-intercepts or different slopes?
Which statement matches a system with no solution: same slope with different $y$-intercepts or different slopes?
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Same slope with different $y$-intercepts. Parallel lines never meet.
Same slope with different $y$-intercepts. Parallel lines never meet.
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Which statement matches infinitely many solutions: same slope with same $y$-intercept or same slope with different $y$-intercepts?
Which statement matches infinitely many solutions: same slope with same $y$-intercept or same slope with different $y$-intercepts?
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Same slope with same $y$-intercept. Same line means all points work.
Same slope with same $y$-intercept. Same line means all points work.
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What does it mean graphically if a system has exactly one solution?
What does it mean graphically if a system has exactly one solution?
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The lines intersect at exactly one point. Two distinct lines cross at a single shared point.
The lines intersect at exactly one point. Two distinct lines cross at a single shared point.
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What does a solution $(x,y)$ to a system of two linear equations represent on their graphs?
What does a solution $(x,y)$ to a system of two linear equations represent on their graphs?
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The intersection point of the two lines at $(x,y)$. Where both lines meet, satisfying both equations simultaneously.
The intersection point of the two lines at $(x,y)$. Where both lines meet, satisfying both equations simultaneously.
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What is the meaning of the ordered pair $(x,y)$ that satisfies both equations in a system?
What is the meaning of the ordered pair $(x,y)$ that satisfies both equations in a system?
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$(x,y)$ makes both equations true at the same time. The point lies on both lines simultaneously.
$(x,y)$ makes both equations true at the same time. The point lies on both lines simultaneously.
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Identify the graphical meaning of the $x$-coordinate of a system's solution point.
Identify the graphical meaning of the $x$-coordinate of a system's solution point.
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The shared $x$-value where the two lines intersect. Both lines pass through this $x$-position.
The shared $x$-value where the two lines intersect. Both lines pass through this $x$-position.
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Identify the graphical meaning of the $y$-coordinate of a system's solution point.
Identify the graphical meaning of the $y$-coordinate of a system's solution point.
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The shared $y$-value where the two lines intersect. Both lines reach this height at intersection.
The shared $y$-value where the two lines intersect. Both lines reach this height at intersection.
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Which option describes the solution set if two lines intersect at $(3,-2)$?
Which option describes the solution set if two lines intersect at $(3,-2)$?
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The system solution is $(3,-2)$. The intersection point is the only solution.
The system solution is $(3,-2)$. The intersection point is the only solution.
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What is the solution of the system $y=2x+1$ and $y=-x+4$?
What is the solution of the system $y=2x+1$ and $y=-x+4$?
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$(1,3)$. Set equations equal: $2x+1=-x+4$, solve for $x=1$.
$(1,3)$. Set equations equal: $2x+1=-x+4$, solve for $x=1$.
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Which relationship between slopes indicates two lines might have no solution?
Which relationship between slopes indicates two lines might have no solution?
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Equal slopes with different $y$-intercepts. Parallel lines never meet when slopes match.
Equal slopes with different $y$-intercepts. Parallel lines never meet when slopes match.
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What does it mean graphically if a system has infinitely many solutions?
What does it mean graphically if a system has infinitely many solutions?
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The lines are the same line and overlap completely. Same slope and $y$-intercept make them identical.
The lines are the same line and overlap completely. Same slope and $y$-intercept make them identical.
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What does it mean graphically if a system has no solution?
What does it mean graphically if a system has no solution?
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The lines are parallel and never intersect. Same slope but different $y$-intercepts prevent crossing.
The lines are parallel and never intersect. Same slope but different $y$-intercepts prevent crossing.
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Which relationship between slopes indicates two lines might have exactly one solution?
Which relationship between slopes indicates two lines might have exactly one solution?
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Different slopes. Non-parallel lines must cross at one point.
Different slopes. Non-parallel lines must cross at one point.
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Which relationship between equations indicates infinitely many solutions?
Which relationship between equations indicates infinitely many solutions?
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One equation is a multiple of the other. Equivalent equations represent the same line.
One equation is a multiple of the other. Equivalent equations represent the same line.
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