Solving Systems of Linear Equations - Algebra
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What is the solution to a system of two linear equations in two variables?
What is the solution to a system of two linear equations in two variables?
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The ordered pair $(x, y)$ that makes both equations true. The point that satisfies both linear equations simultaneously.
The ordered pair $(x, y)$ that makes both equations true. The point that satisfies both linear equations simultaneously.
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What method solves a system by solving one equation for a variable and substituting?
What method solves a system by solving one equation for a variable and substituting?
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The substitution method. Replace one variable with an expression from the other equation.
The substitution method. Replace one variable with an expression from the other equation.
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Identify the solution: Solve the system $2x+y=9$ and $x-y=3$.
Identify the solution: Solve the system $2x+y=9$ and $x-y=3$.
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$(4,1)$. Add equations to eliminate $y$: $3x=12$, so $x=4$.
$(4,1)$. Add equations to eliminate $y$: $3x=12$, so $x=4$.
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Identify the solution type: Solve the system $2x+3y=12$ and $4x+6y=10$.
Identify the solution type: Solve the system $2x+3y=12$ and $4x+6y=10$.
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No solution. The second equation simplifies to $2x+3y=5$, contradicting the first.
No solution. The second equation simplifies to $2x+3y=5$, contradicting the first.
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Identify the solution type: Solve the system $x+y=7$ and $x+y=9$.
Identify the solution type: Solve the system $x+y=7$ and $x+y=9$.
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No solution. Same left sides but different right sides creates a contradiction.
No solution. Same left sides but different right sides creates a contradiction.
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Identify the solution: Solve the system $y=3x-2$ and $y=3x+5$.
Identify the solution: Solve the system $y=3x-2$ and $y=3x+5$.
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No solution. Same slope but different y-intercepts means parallel lines.
No solution. Same slope but different y-intercepts means parallel lines.
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Identify the solution: Solve the system $x= -3$ and $y=5$.
Identify the solution: Solve the system $x= -3$ and $y=5$.
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$(-3,5)$. Both variables are directly given as constants.
$(-3,5)$. Both variables are directly given as constants.
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Identify the solution: Solve the system $x+y=1$ and $x=4$.
Identify the solution: Solve the system $x+y=1$ and $x=4$.
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$(4,-3)$. Substitute $x=4$ into first equation to get $y=-3$.
$(4,-3)$. Substitute $x=4$ into first equation to get $y=-3$.
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Identify the solution: Solve the system $2x-y=7$ and $x+y=5$.
Identify the solution: Solve the system $2x-y=7$ and $x+y=5$.
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$(4,1)$. Add equations to eliminate $y$: $3x=12$, so $x=4$.
$(4,1)$. Add equations to eliminate $y$: $3x=12$, so $x=4$.
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Identify the solution: Solve the system $x+3y=3$ and $x=3$.
Identify the solution: Solve the system $x+3y=3$ and $x=3$.
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$(3,0)$. Substitute $x=3$ into first equation to get $y=0$.
$(3,0)$. Substitute $x=3$ into first equation to get $y=0$.
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Identify the solution: Solve the system $x+y=0$ and $y=2x$.
Identify the solution: Solve the system $x+y=0$ and $y=2x$.
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$(0,0)$. Substitute $y=2x$ into first equation: $x+2x=0$.
$(0,0)$. Substitute $y=2x$ into first equation: $x+2x=0$.
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Identify the solution: Solve the system $3x-3y=6$ and $x-y=2$.
Identify the solution: Solve the system $3x-3y=6$ and $x-y=2$.
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Infinitely many solutions. The second equation is equivalent to the first when simplified.
Infinitely many solutions. The second equation is equivalent to the first when simplified.
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Identify the solution: Solve the system $y=2x$ and $y=2x+6$.
Identify the solution: Solve the system $y=2x$ and $y=2x+6$.
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No solution. Same slope but different y-intercepts means parallel lines.
No solution. Same slope but different y-intercepts means parallel lines.
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Identify the solution: Solve the system $x+y=9$ and $y=7-x$.
Identify the solution: Solve the system $x+y=9$ and $y=7-x$.
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No solution. Different y-intercepts with same slope means parallel lines.
No solution. Different y-intercepts with same slope means parallel lines.
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Identify the solution: Solve the system $y=4x$ and $y=4x-8$.
Identify the solution: Solve the system $y=4x$ and $y=4x-8$.
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No solution. Same slope but different y-intercepts means parallel lines.
No solution. Same slope but different y-intercepts means parallel lines.
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What method solves a system by graphing and finding the intersection point?
What method solves a system by graphing and finding the intersection point?
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The graphing method. Plot both lines and find where they cross.
The graphing method. Plot both lines and find where they cross.
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Identify the solution: Solve the system $x+y=5$ and $x+2y=8$.
Identify the solution: Solve the system $x+y=5$ and $x+2y=8$.
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$(2,3)$. Subtract equations to eliminate $x$: $-y=-3$, so $y=3$.
$(2,3)$. Subtract equations to eliminate $x$: $-y=-3$, so $y=3$.
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Identify the solution: Solve the system $x+y=7$ and $2x+2y=14$.
Identify the solution: Solve the system $x+y=7$ and $2x+2y=14$.
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Infinitely many solutions. The second equation is twice the first, so they're equivalent.
Infinitely many solutions. The second equation is twice the first, so they're equivalent.
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Identify the solution: Solve the system $2x+y=9$ and $x-y=3$.
Identify the solution: Solve the system $2x+y=9$ and $x-y=3$.
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$(4,1)$. Add equations to eliminate $y$: $3x=12$, so $x=4$.
$(4,1)$. Add equations to eliminate $y$: $3x=12$, so $x=4$.
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Identify the solution: Solve the system $x+2y=8$ and $x=2$.
Identify the solution: Solve the system $x+2y=8$ and $x=2$.
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$(2,3)$. Substitute $x=2$ into first equation to get $y=3$.
$(2,3)$. Substitute $x=2$ into first equation to get $y=3$.
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Identify the solution: Solve the system $y=-x+4$ and $y=x$.
Identify the solution: Solve the system $y=-x+4$ and $y=x$.
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$(2,2)$. Set equations equal: $-x+4=x$ gives $x=2$.
$(2,2)$. Set equations equal: $-x+4=x$ gives $x=2$.
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Identify the solution: Solve the system $3x+y=12$ and $y=3$.
Identify the solution: Solve the system $3x+y=12$ and $y=3$.
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$(3,3)$. Substitute $y=3$ into first equation to get $x=3$.
$(3,3)$. Substitute $y=3$ into first equation to get $x=3$.
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Identify the solution: Solve the system $2x+3y=12$ and $2x+3y=12$.
Identify the solution: Solve the system $2x+3y=12$ and $2x+3y=12$.
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Infinitely many solutions. Both equations are identical, so all points satisfy both.
Infinitely many solutions. Both equations are identical, so all points satisfy both.
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Identify the solution type: Solve the system $2x+3y=12$ and $4x+6y=10$.
Identify the solution type: Solve the system $2x+3y=12$ and $4x+6y=10$.
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No solution. The second equation simplifies to $2x+3y=5$, contradicting the first.
No solution. The second equation simplifies to $2x+3y=5$, contradicting the first.
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Identify the solution: Solve the system $x=1$ and $2x+y=10$.
Identify the solution: Solve the system $x=1$ and $2x+y=10$.
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$(1,8)$. Substitute $x=1$ into second equation to get $y=8$.
$(1,8)$. Substitute $x=1$ into second equation to get $y=8$.
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Identify the solution: Solve the system $2x+3y=13$ and $y=3$.
Identify the solution: Solve the system $2x+3y=13$ and $y=3$.
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$(2,3)$. Substitute $y=3$ into first equation to get $x=2$.
$(2,3)$. Substitute $y=3$ into first equation to get $x=2$.
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Identify the solution: Solve the system $y=\frac{1}{2}x+2$ and $y=-\frac{1}{2}x+4$.
Identify the solution: Solve the system $y=\frac{1}{2}x+2$ and $y=-\frac{1}{2}x+4$.
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$(2,3)$. Set equations equal: $\frac{1}{2}x+2=-\frac{1}{2}x+4$ gives $x=2$.
$(2,3)$. Set equations equal: $\frac{1}{2}x+2=-\frac{1}{2}x+4$ gives $x=2$.
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Identify the solution: Solve the system $x+y=6$ and $y=2$.
Identify the solution: Solve the system $x+y=6$ and $y=2$.
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$(4,2)$. Substitute $y=2$ into first equation to get $x=4$.
$(4,2)$. Substitute $y=2$ into first equation to get $x=4$.
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Identify the solution: Solve the system $x+4y=16$ and $x+4y=0$.
Identify the solution: Solve the system $x+4y=16$ and $x+4y=0$.
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No solution. Same left sides but different right sides creates a contradiction.
No solution. Same left sides but different right sides creates a contradiction.
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Identify the solution: Solve the system $2x-y=4$ and $y=2x-4$.
Identify the solution: Solve the system $2x-y=4$ and $y=2x-4$.
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Infinitely many solutions. Both equations represent the same line when rearranged.
Infinitely many solutions. Both equations represent the same line when rearranged.
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