Solving Systems of Linear Equations - Algebra 2
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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 lines coincide: equivalent equations for the same line. Every point on the line satisfies both equations.
The lines coincide: equivalent equations for the same line. Every point on the line satisfies both equations.
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What is the slope-intercept form of a linear equation?
What is the slope-intercept form of a linear equation?
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$y=mx+b$. Standard form showing slope $m$ and $y$-intercept $b$.
$y=mx+b$. Standard form showing slope $m$ and $y$-intercept $b$.
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What is the standard form of a linear equation used for elimination?
What is the standard form of a linear equation used for elimination?
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$Ax+By=C$. Format ideal for elimination method alignment.
$Ax+By=C$. Format ideal for elimination method alignment.
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Identify the condition for the same line in $y=mx+b$ form.
Identify the condition for the same line in $y=mx+b$ form.
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Equal slopes and intercepts: $m_1=m_2$ and $b_1=b_2$. Identical lines when both parameters match.
Equal slopes and intercepts: $m_1=m_2$ and $b_1=b_2$. Identical lines when both parameters match.
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What is the solution of the system $y=\frac{1}{2}x+3$ and $y=-\frac{1}{2}x+1$?
What is the solution of the system $y=\frac{1}{2}x+3$ and $y=-\frac{1}{2}x+1$?
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$(-2,2)$. Set slopes equal: $\frac{1}{2}x+3=-\frac{1}{2}x+1$, solve.
$(-2,2)$. Set slopes equal: $\frac{1}{2}x+3=-\frac{1}{2}x+1$, solve.
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What is the solution of the system $y=3x-5$ and $2y=6x-10$?
What is the solution of the system $y=3x-5$ and $2y=6x-10$?
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Infinitely many solutions (same line). Second equation is first multiplied by 2.
Infinitely many solutions (same line). Second equation is first multiplied by 2.
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What is the solution of the system $y=3x-5$ and $2y=6x-10$?
What is the solution of the system $y=3x-5$ and $2y=6x-10$?
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Infinitely many solutions (same line). Second equation is first multiplied by 2.
Infinitely many solutions (same line). Second equation is first multiplied by 2.
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What is the solution type for $y=2x+1$ and $y=2x-3$?
What is the solution type for $y=2x+1$ and $y=2x-3$?
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No solution (parallel lines). Same slope $m=2$, different intercepts.
No solution (parallel lines). Same slope $m=2$, different intercepts.
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What is the solution type for $2x+4y=8$ and $x+2y=4$?
What is the solution type for $2x+4y=8$ and $x+2y=4$?
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Infinitely many solutions (equivalent equations). Second equation is first divided by 2.
Infinitely many solutions (equivalent equations). Second equation is first divided by 2.
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What is the solution type for $3x-6y=9$ and $x-2y=5$?
What is the solution type for $3x-6y=9$ and $x-2y=5$?
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No solution (parallel lines). Same slope $m=\frac{3}{2}$, different intercepts.
No solution (parallel lines). Same slope $m=\frac{3}{2}$, different intercepts.
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What method solves a system by making coefficients opposites and adding equations?
What method solves a system by making coefficients opposites and adding equations?
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Elimination (linear combination). Add equations after making coefficients opposites.
Elimination (linear combination). Add equations after making coefficients opposites.
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What is the standard form of a linear equation used for elimination?
What is the standard form of a linear equation used for elimination?
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$Ax+By=C$. Format ideal for elimination method alignment.
$Ax+By=C$. Format ideal for elimination method alignment.
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What method solves a system by making coefficients opposites and adding equations?
What method solves a system by making coefficients opposites and adding equations?
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Elimination (linear combination). Add equations after making coefficients opposites.
Elimination (linear combination). Add equations after making coefficients opposites.
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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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Substitution. Replace one variable with expression from other equation.
Substitution. Replace one variable with expression from other equation.
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What method solves a system by finding the intersection point of two graphed lines?
What method solves a system by finding the intersection point of two graphed lines?
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Graphing (approximate intersection). Visual method showing where lines cross.
Graphing (approximate intersection). Visual method showing where lines cross.
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Identify the condition for parallel lines in $y=mx+b$ form.
Identify the condition for parallel lines in $y=mx+b$ form.
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Equal slopes $m_1=m_2$ and different intercepts $b_1\ne b_2$. Lines never meet when slopes match but intercepts differ.
Equal slopes $m_1=m_2$ and different intercepts $b_1\ne b_2$. Lines never meet when slopes match but intercepts differ.
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Identify the system solution type if elimination gives a true statement like $0=0$.
Identify the system solution type if elimination gives a true statement like $0=0$.
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Infinitely many solutions. Variables cancel to give identity $0=0$.
Infinitely many solutions. Variables cancel to give identity $0=0$.
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Identify the system solution type if elimination gives a false statement like $0=5$.
Identify the system solution type if elimination gives a false statement like $0=5$.
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No solution. Variables cancel to give contradiction.
No solution. Variables cancel to give contradiction.
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Identify the condition for the same line in $y=mx+b$ form.
Identify the condition for the same line in $y=mx+b$ form.
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Equal slopes and intercepts: $m_1=m_2$ and $b_1=b_2$. Identical lines when both parameters match.
Equal slopes and intercepts: $m_1=m_2$ and $b_1=b_2$. Identical lines when both parameters match.
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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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$(3,7)$. Set equations equal: $2x+1=x+4$, solve for $x=3$.
$(3,7)$. Set equations equal: $2x+1=x+4$, solve for $x=3$.
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What is the slope of the line $3x-6y=12$?
What is the slope of the line $3x-6y=12$?
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$\frac{1}{2}$. Rewrite as $y=\frac{1}{2}x-2$ to find slope.
$\frac{1}{2}$. Rewrite as $y=\frac{1}{2}x-2$ to find slope.
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What is the solution of the system $y=-x+2$ and $y=x-4$?
What is the solution of the system $y=-x+2$ and $y=x-4$?
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$(3,-1)$. Set equations equal: $-x+2=x-4$, solve for $x=3$.
$(3,-1)$. Set equations equal: $-x+2=x-4$, solve for $x=3$.
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What is the solution of the system $3x-2y=4$ and $x+2y=8$?
What is the solution of the system $3x-2y=4$ and $x+2y=8$?
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$(3,\frac{5}{2})$. Add equations to eliminate $y$: $4x=12$, so $x=3$.
$(3,\frac{5}{2})$. Add equations to eliminate $y$: $4x=12$, so $x=3$.
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What is the solution of the system $x+y=10$ and $x-y=2$?
What is the solution of the system $x+y=10$ and $x-y=2$?
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$(6,4)$. Add equations to get $2x=12$, so $x=6$.
$(6,4)$. Add equations to get $2x=12$, so $x=6$.
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What is the solution of the system $4x-y=9$ and $2x+y=3$?
What is the solution of the system $4x-y=9$ and $2x+y=3$?
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$(2,-1)$. Add equations to eliminate $y$: $6x=12$, so $x=2$.
$(2,-1)$. Add equations to eliminate $y$: $6x=12$, so $x=2$.
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What is the solution of the system $5x+y=1$ and $x-y=7$?
What is the solution of the system $5x+y=1$ and $x-y=7$?
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$(\frac{4}{3},-\frac{17}{3})$. Add equations to eliminate $y$: $6x=8$, so $x=\frac{4}{3}$.
$(\frac{4}{3},-\frac{17}{3})$. Add equations to eliminate $y$: $6x=8$, so $x=\frac{4}{3}$.
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What is the solution of the system $y=\frac{1}{2}x+3$ and $y=-\frac{1}{2}x+1$?
What is the solution of the system $y=\frac{1}{2}x+3$ and $y=-\frac{1}{2}x+1$?
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$(-2,2)$. Set slopes equal: $\frac{1}{2}x+3=-\frac{1}{2}x+1$, solve.
$(-2,2)$. Set slopes equal: $\frac{1}{2}x+3=-\frac{1}{2}x+1$, solve.
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What is the solution of the system $x=4$ and $y=-2x+1$?
What is the solution of the system $x=4$ and $y=-2x+1$?
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$(4,-7)$. Substitute $x=4$ into second equation.
$(4,-7)$. Substitute $x=4$ into second equation.
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What is the solution of the system $y=6$ and $2x+y=10$?
What is the solution of the system $y=6$ and $2x+y=10$?
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$(2,6)$. Substitute $y=6$ into second equation.
$(2,6)$. Substitute $y=6$ into second equation.
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What is the solution of the system $x+y=5$ and $2x+2y=8$?
What is the solution of the system $x+y=5$ and $2x+2y=8$?
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No solution. Second equation gives $x+y=4$, contradicting first.
No solution. Second equation gives $x+y=4$, contradicting first.
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