Solve Real-World System Problems - 8th Grade Math
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What is the solution type for the system: $2x+4y=10$ and $x+2y=5$?
What is the solution type for the system: $2x+4y=10$ and $x+2y=5$?
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Infinitely many solutions. Second equation is half the first, so they're the same line.
Infinitely many solutions. Second equation is half the first, so they're the same line.
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What system models: $x$ adult tickets and $y$ student tickets, $x+y=30$, adults $8$ dollars, students $5$ dollars, total $201$ dollars?
What system models: $x$ adult tickets and $y$ student tickets, $x+y=30$, adults $8$ dollars, students $5$ dollars, total $201$ dollars?
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$x+y=30$ and $8x+5y=201$. First equation: total tickets; second: total revenue.
$x+y=30$ and $8x+5y=201$. First equation: total tickets; second: total revenue.
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Identify the solution to the system: $3x-2y=4$ and $x-y=2$.
Identify the solution to the system: $3x-2y=4$ and $x-y=2$.
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$(0,-2)$. From second equation $x=y+2$; substitute to get $y=-2$, $x=0$.
$(0,-2)$. From second equation $x=y+2$; substitute to get $y=-2$, $x=0$.
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What does it mean if a system of two linear equations has infinitely many solutions?
What does it mean if a system of two linear equations has infinitely many solutions?
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The equations represent the same line. Both equations simplify to the same slope and y-intercept.
The equations represent the same line. Both equations simplify to the same slope and y-intercept.
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Identify the solution to the system: $2x+y=9$ and $x+y=6$.
Identify the solution to the system: $2x+y=9$ and $x+y=6$.
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$(3,3)$. Subtract second from first: $x=3$; substitute to get $y=3$.
$(3,3)$. Subtract second from first: $x=3$; substitute to get $y=3$.
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What is the substitution method for solving a system of linear equations?
What is the substitution method for solving a system of linear equations?
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Solve one equation for a variable, substitute into the other. This reduces the system to one equation with one variable.
Solve one equation for a variable, substitute into the other. This reduces the system to one equation with one variable.
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Identify the solution to the system: $x+2y=10$ and $x-y=1$.
Identify the solution to the system: $x+2y=10$ and $x-y=1$.
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$(4,3)$. From second equation $x=y+1$; substitute to get $y=3$, $x=4$.
$(4,3)$. From second equation $x=y+1$; substitute to get $y=3$, $x=4$.
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What is the elimination method for solving a system of linear equations?
What is the elimination method for solving a system of linear equations?
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Add or subtract equations to eliminate one variable. Combining equations strategically simplifies to one variable.
Add or subtract equations to eliminate one variable. Combining equations strategically simplifies to one variable.
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Identify the solution to the system: $2x+3y=12$ and $x+3y=9$.
Identify the solution to the system: $2x+3y=12$ and $x+3y=9$.
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$(3,2)$. Subtract second from first: $x=3$; substitute to get $y=2$.
$(3,2)$. Subtract second from first: $x=3$; substitute to get $y=2$.
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What is the first step to solve a word problem with a system of equations?
What is the first step to solve a word problem with a system of equations?
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Define variables for the unknown quantities. Variables represent what you're solving for.
Define variables for the unknown quantities. Variables represent what you're solving for.
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Which form is most useful to graph quickly: standard form $Ax+By=C$ or slope-intercept $y=mx+b$?
Which form is most useful to graph quickly: standard form $Ax+By=C$ or slope-intercept $y=mx+b$?
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Slope-intercept form $y=mx+b$. Shows slope and y-intercept directly for easy graphing.
Slope-intercept form $y=mx+b$. Shows slope and y-intercept directly for easy graphing.
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Identify whether the system has $0,1,$ or infinitely many solutions: $y=2x+3$ and $y=2x-1$.
Identify whether the system has $0,1,$ or infinitely many solutions: $y=2x+3$ and $y=2x-1$.
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$0$ solutions. Same slope (2) but different y-intercepts means parallel lines.
$0$ solutions. Same slope (2) but different y-intercepts means parallel lines.
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Identify whether the system has $0,1,$ or infinitely many solutions: $2x+4y=8$ and $x+2y=4$.
Identify whether the system has $0,1,$ or infinitely many solutions: $2x+4y=8$ and $x+2y=4$.
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Infinitely many solutions. Second equation is half the first; they're the same line.
Infinitely many solutions. Second equation is half the first; they're the same line.
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What does it mean for an ordered pair $(x,y)$ to be a solution to a system of equations?
What does it mean for an ordered pair $(x,y)$ to be a solution to a system of equations?
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$(x,y)$ makes both equations true when substituted. Both equations must be satisfied simultaneously.
$(x,y)$ makes both equations true when substituted. Both equations must be satisfied simultaneously.
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Identify whether the system has $0,1,$ or infinitely many solutions: $y=-x+5$ and $y=2x-1$.
Identify whether the system has $0,1,$ or infinitely many solutions: $y=-x+5$ and $y=2x-1$.
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$1$ solution. Different slopes (-1 and 2) guarantee one intersection.
$1$ solution. Different slopes (-1 and 2) guarantee one intersection.
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What does it mean if a system of two linear equations has no solution?
What does it mean if a system of two linear equations has no solution?
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The lines are parallel and never intersect. Parallel lines have the same slope but different y-intercepts.
The lines are parallel and never intersect. Parallel lines have the same slope but different y-intercepts.
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What does it mean if a system of two linear equations has exactly one solution?
What does it mean if a system of two linear equations has exactly one solution?
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The lines intersect at exactly one point. The unique intersection point is the solution.
The lines intersect at exactly one point. The unique intersection point is the solution.
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Identify the meaning of the solution $(x,y)$ in a cost problem with equations for total items and total cost.
Identify the meaning of the solution $(x,y)$ in a cost problem with equations for total items and total cost.
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$(x,y)$ gives the quantities of the two item types. Solution tells how many of each item type was purchased.
$(x,y)$ gives the quantities of the two item types. Solution tells how many of each item type was purchased.
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What system models: perimeter $50$ of a rectangle with length $x$ and width $y$, and $x=y+5$?
What system models: perimeter $50$ of a rectangle with length $x$ and width $y$, and $x=y+5$?
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$2x+2y=50$ and $x=y+5$. Perimeter formula and length-width relationship.
$2x+2y=50$ and $x=y+5$. Perimeter formula and length-width relationship.
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What condition indicates a system has infinitely many solutions (dependent)?
What condition indicates a system has infinitely many solutions (dependent)?
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Equations are equivalent (same line)
Equations are equivalent (same line)
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What condition on slopes indicates a system has no solution (inconsistent)?
What condition on slopes indicates a system has no solution (inconsistent)?
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Same slope, different $y$-intercepts
Same slope, different $y$-intercepts
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What are the three possible numbers of solutions for two linear equations in two variables?
What are the three possible numbers of solutions for two linear equations in two variables?
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$0$ solutions, $1$ solution, or infinitely many solutions
$0$ solutions, $1$ solution, or infinitely many solutions
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What does it mean for an ordered pair $(x, y)$ to solve a system of two equations?
What does it mean for an ordered pair $(x, y)$ to solve a system of two equations?
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$(x, y)$ makes both equations true at the same time
$(x, y)$ makes both equations true at the same time
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Solve the system: $3x+2y=12$ and $x+2y=4$.
Solve the system: $3x+2y=12$ and $x+2y=4$.
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$(4,0)$
$(4,0)$
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Solve the system: $2x+y=11$ and $x-y=1$.
Solve the system: $2x+y=11$ and $x-y=1$.
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$(4,3)$
$(4,3)$
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Solve the system by elimination: $x+y=9$ and $x-y=1$.
Solve the system by elimination: $x+y=9$ and $x-y=1$.
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$(5,4)$
$(5,4)$
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What does a negative solution like $(x,y)=(-2,5)$ indicate in a real-world context?
What does a negative solution like $(x,y)=(-2,5)$ indicate in a real-world context?
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Check context; it may be invalid if quantities cannot be negative
Check context; it may be invalid if quantities cannot be negative
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Identify the solution type: $y=2x+3$ and $y=2x-5$.
Identify the solution type: $y=2x+3$ and $y=2x-5$.
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No solution
No solution
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Solve the system: $y=x-2$ and $y=-2x+7$.
Solve the system: $y=x-2$ and $y=-2x+7$.
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$(3,1)$
$(3,1)$
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Identify the solution type: $3x-6y=9$ and $x-2y=3$.
Identify the solution type: $3x-6y=9$ and $x-2y=3$.
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Infinitely many solutions
Infinitely many solutions
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