Systems of Equations - PSAT Math
Card 1 of 30
What is the solution to the system $3x-2y=4$ and $x+2y=8$?
What is the solution to 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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Identify the solution count for $2x+4y=8$ and $x+2y=5$.
Identify the solution count for $2x+4y=8$ and $x+2y=5$.
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No solution. Second equation is $2x+4y=10$, parallel to first.
No solution. Second equation is $2x+4y=10$, parallel to first.
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Identify the solution count for $2x+4y=8$ and $x+2y=4$.
Identify the solution count for $2x+4y=8$ and $x+2y=4$.
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Infinitely many solutions. Second equation is half the first, same line.
Infinitely many solutions. Second equation is half the first, same line.
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What is the value of $k$ that makes the system $x+ky=6$ and $2x+4y=12$ have infinitely many solutions?
What is the value of $k$ that makes the system $x+ky=6$ and $2x+4y=12$ have infinitely many solutions?
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$k=2$. For same line, coefficients must be proportional: $\frac{1}{2}=\frac{k}{4}$.
$k=2$. For same line, coefficients must be proportional: $\frac{1}{2}=\frac{k}{4}$.
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What is the solution to the system $y=x^2$ and $y=4$?
What is the solution to the system $y=x^2$ and $y=4$?
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$(2,4)$ and $(-2,4)$. Substitute $y=4$ into $y=x^2$ to get $x^2=4$.
$(2,4)$ and $(-2,4)$. Substitute $y=4$ into $y=x^2$ to get $x^2=4$.
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What is the solution to the system $y=x^2$ and $y=x+2$?
What is the solution to the system $y=x^2$ and $y=x+2$?
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$(-1,1)$ and $(2,4)$. Set $x^2=x+2$, solve quadratic $x^2-x-2=0$.
$(-1,1)$ and $(2,4)$. Set $x^2=x+2$, solve quadratic $x^2-x-2=0$.
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What is the solution to the system $x^2+y^2=25$ and $y=0$?
What is the solution to the system $x^2+y^2=25$ and $y=0$?
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$(5,0)$ and $(-5,0)$. Circle intersects x-axis where $y=0$, so $x^2=25$.
$(5,0)$ and $(-5,0)$. Circle intersects x-axis where $y=0$, so $x^2=25$.
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What is the meaning of a solution to a system of equations?
What is the meaning of a solution to a system of equations?
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An ordered pair $(x,y)$ that makes both equations true. Values that satisfy both equations simultaneously.
An ordered pair $(x,y)$ that makes both equations true. Values that satisfy both equations simultaneously.
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Identify the solution count if $a_1b_2-a_2b_1=0$ for two linear equations in $x$ and $y$.
Identify the solution count if $a_1b_2-a_2b_1=0$ for two linear equations in $x$ and $y$.
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Either no solution or infinitely many solutions. Zero determinant means parallel or same line.
Either no solution or infinitely many solutions. Zero determinant means parallel or same line.
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What is the determinant test for a unique solution to $a_1x+b_1y=c_1$ and $a_2x+b_2y=c_2$?
What is the determinant test for a unique solution to $a_1x+b_1y=c_1$ and $a_2x+b_2y=c_2$?
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Unique solution if $a_1b_2-a_2b_1\ne 0$. Non-zero determinant means lines intersect at one point.
Unique solution if $a_1b_2-a_2b_1\ne 0$. Non-zero determinant means lines intersect at one point.
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What is the slope-intercept form you may graph to solve a linear system?
What is the slope-intercept form you may graph to solve a linear system?
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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 solution to the system $y=2x+1$ and $y=5$?
What is the solution to the system $y=2x+1$ and $y=5$?
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$(2,5)$. Substitute $y=5$ into first equation: $5=2x+1$, so $x=2$.
$(2,5)$. Substitute $y=5$ into first equation: $5=2x+1$, so $x=2$.
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What is the solution to the system $x+y=7$ and $x-y=1$?
What is the solution to the system $x+y=7$ and $x-y=1$?
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$(4,3)$. Add equations to get $2x=8$, so $x=4$ and $y=3$.
$(4,3)$. Add equations to get $2x=8$, so $x=4$ and $y=3$.
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What is the solution to the system $2x+y=9$ and $x-y=1$?
What is the solution to the system $2x+y=9$ and $x-y=1$?
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$(\frac{10}{3},\frac{7}{3})$. Add equations to get $3x=10$, then substitute back.
$(\frac{10}{3},\frac{7}{3})$. Add equations to get $3x=10$, then substitute back.
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What does the intersection point of two lines represent in a linear system?
What does the intersection point of two lines represent in a linear system?
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The solution $(x,y)$ that satisfies both equations. The common point where both lines meet.
The solution $(x,y)$ that satisfies both equations. The common point where both lines meet.
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What does it mean if two equations in a linear system represent the same line?
What does it mean if two equations in a linear system represent the same line?
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Infinitely many solutions. Same line means every point on it satisfies both equations.
Infinitely many solutions. Same line means every point on it satisfies both equations.
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What are the three possible solution counts for a system of two linear equations in two variables?
What are the three possible solution counts for a system of two linear equations in two variables?
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One solution, no solution, or infinitely many solutions. Linear systems can intersect once, never, or everywhere.
One solution, no solution, or infinitely many solutions. Linear systems can intersect once, never, or everywhere.
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What does it mean if two lines in a linear system are parallel and distinct?
What does it mean if two lines in a linear system are parallel and distinct?
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No solution. Parallel lines never intersect, so no common point exists.
No solution. Parallel lines never intersect, so no common point exists.
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What is the substitution method for solving a system used for?
What is the substitution method for solving a system used for?
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Replacing a variable using one equation to solve the other. Express one variable in terms of the other, then substitute.
Replacing a variable using one equation to solve the other. Express one variable in terms of the other, then substitute.
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What is the elimination method for solving a system used for?
What is the elimination method for solving a system used for?
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Adding or subtracting equations to eliminate a variable. Combine equations to cancel out one variable.
Adding or subtracting equations to eliminate a variable. Combine equations to cancel out one variable.
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Identify the solution to the system: $2x+y=9$ and $x-y=0$.
Identify the solution to the system: $2x+y=9$ and $x-y=0$.
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$(3,3)$. From $x-y=0$, we get $x=y$; substituting gives $3x=9$.
$(3,3)$. From $x-y=0$, we get $x=y$; substituting gives $3x=9$.
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What condition on slopes indicates a system of two lines has no solution?
What condition on slopes indicates a system of two lines has no solution?
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Same slope, different $y$-intercepts. Parallel lines never intersect.
Same slope, different $y$-intercepts. Parallel lines never intersect.
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What are the three possible numbers of solutions for a system of two linear equations?
What are the three possible numbers of solutions for a system of two linear equations?
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$0$, $1$, or infinitely many solutions. Linear systems can have unique intersection, parallel lines, or identical lines.
$0$, $1$, or infinitely many solutions. Linear systems can have unique intersection, parallel lines, or identical lines.
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What is the quickest way to choose between substitution and elimination?
What is the quickest way to choose between substitution and elimination?
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Use substitution if a variable is isolated; otherwise use elimination. Isolated variables make substitution faster than elimination.
Use substitution if a variable is isolated; otherwise use elimination. Isolated variables make substitution faster than elimination.
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Identify the solution to the system: $x+y=7$ and $x-y=1$.
Identify the solution to the system: $x+y=7$ and $x-y=1$.
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$(4,3)$. Adding equations gives $2x=8$, so $x=4$ and $y=3$.
$(4,3)$. Adding equations gives $2x=8$, so $x=4$ and $y=3$.
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Identify the solution to the system: $y=2x+1$ and $y=9$.
Identify the solution to the system: $y=2x+1$ and $y=9$.
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$(4,9)$. Substitute $y=9$ into first equation: $9=2x+1$, so $x=4$.
$(4,9)$. Substitute $y=9$ into first equation: $9=2x+1$, so $x=4$.
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Identify the solution to the system: $y=-x+6$ and $y=x$.
Identify the solution to the system: $y=-x+6$ and $y=x$.
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$(3,3)$. Setting equations equal: $-x+6=x$ gives $x=3$, so $y=3$.
$(3,3)$. Setting equations equal: $-x+6=x$ gives $x=3$, so $y=3$.
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What is the solution type for the system: $y=2x+3$ and $y=2x-5$?
What is the solution type for the system: $y=2x+3$ and $y=2x-5$?
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No solution. Same slope $m=2$ but different $y$-intercepts means parallel lines.
No solution. Same slope $m=2$ but different $y$-intercepts means parallel lines.
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What is the solution type for the system: $2x+4y=8$ and $x+2y=4$?
What is the solution type for the system: $2x+4y=8$ and $x+2y=4$?
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Infinitely many solutions. Second equation is first divided by 2, so they're the same line.
Infinitely many solutions. Second equation is first divided by 2, so they're the same line.
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What is the solution type if elimination produces a true statement like $0=0$?
What is the solution type if elimination produces a true statement like $0=0$?
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Infinitely many solutions. All variables cancel, leaving a true identity.
Infinitely many solutions. All variables cancel, leaving a true identity.
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