Graphs as Sets of Solutions - Algebra
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Which ordered pair is on the graph of $x=5$ : $(5,1)$ or $(1,5)$?
Which ordered pair is on the graph of $x=5$ : $(5,1)$ or $(1,5)$?
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$(5,1)$. Vertical line equations have form $x = k$.
$(5,1)$. Vertical line equations have form $x = k$.
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What does it mean if a point $(x,y)$ lies on the graph of an equation?
What does it mean if a point $(x,y)$ lies on the graph of an equation?
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$(x,y)$ is a solution; it makes the equation true when substituted. Points on the graph satisfy the equation when coordinates are substituted.
$(x,y)$ is a solution; it makes the equation true when substituted. Points on the graph satisfy the equation when coordinates are substituted.
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What is the graph of an equation in two variables defined to be?
What is the graph of an equation in two variables defined to be?
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The set of all $(x,y)$ solutions plotted in the coordinate plane. A graph visualizes every solution as points in the plane.
The set of all $(x,y)$ solutions plotted in the coordinate plane. A graph visualizes every solution as points in the plane.
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What does the equation $x=k$ (constant) graph as?
What does the equation $x=k$ (constant) graph as?
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A vertical line at $x=k$. All points have the same $x$-coordinate value.
A vertical line at $x=k$. All points have the same $x$-coordinate value.
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Which equation has a graph that is a line: $y=2x+1$ or $y=x^2$?
Which equation has a graph that is a line: $y=2x+1$ or $y=x^2$?
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$y=2x+1$. Linear equations graph as straight lines, quadratics as curves.
$y=2x+1$. Linear equations graph as straight lines, quadratics as curves.
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What set of points is described by the equation $x=0$?
What set of points is described by the equation $x=0$?
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All points on the $y$-axis: $(0,y)$. The $y$-axis is defined by $x = 0$.
All points on the $y$-axis: $(0,y)$. The $y$-axis is defined by $x = 0$.
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Identify the correct interpretation of $x+y=2$: does it describe points or just $x$-values?
Identify the correct interpretation of $x+y=2$: does it describe points or just $x$-values?
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It describes points $(x,y)$ whose sum is $2$. Equations in two variables describe coordinate relationships.
It describes points $(x,y)$ whose sum is $2$. Equations in two variables describe coordinate relationships.
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What set of points is described by the equation $y=0$?
What set of points is described by the equation $y=0$?
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All points on the $x$-axis: $(x,0)$. The $x$-axis is defined by $y = 0$.
All points on the $x$-axis: $(x,0)$. The $x$-axis is defined by $y = 0$.
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Identify whether $(2,2)$ is on the graph of $y=\frac{1}{2}x+\frac{3}{2}$.
Identify whether $(2,2)$ is on the graph of $y=\frac{1}{2}x+\frac{3}{2}$.
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No, because $2\ne \frac{1}{2}(2)+\frac{3}{2}$. Substitution shows: $2 \neq \frac{1}{2}(2) + \frac{3}{2} = \frac{5}{2}$.
No, because $2\ne \frac{1}{2}(2)+\frac{3}{2}$. Substitution shows: $2 \neq \frac{1}{2}(2) + \frac{3}{2} = \frac{5}{2}$.
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Identify whether $(1,2)$ is on the graph of $y=\frac{1}{2}x+\frac{3}{2}$.
Identify whether $(1,2)$ is on the graph of $y=\frac{1}{2}x+\frac{3}{2}$.
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Yes, because $2=\frac{1}{2}(1)+\frac{3}{2}$. Substitution confirms: $2 = \frac{1}{2}(1) + \frac{3}{2} = 2$.
Yes, because $2=\frac{1}{2}(1)+\frac{3}{2}$. Substitution confirms: $2 = \frac{1}{2}(1) + \frac{3}{2} = 2$.
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Identify the error: "The graph of $y=2x$ is all $x$ values that work." What should it say?
Identify the error: "The graph of $y=2x$ is all $x$ values that work." What should it say?
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The graph is all ordered pairs $(x,y)$ that satisfy $y=2x$. Graphs represent ordered pairs, not just individual variable values.
The graph is all ordered pairs $(x,y)$ that satisfy $y=2x$. Graphs represent ordered pairs, not just individual variable values.
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Which equation has a graph that is a curve: $x+y=4$ or $y=x^2+1$?
Which equation has a graph that is a curve: $x+y=4$ or $y=x^2+1$?
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$y=x^2+1$. Quadratic equations graph as parabolas, linear equations as lines.
$y=x^2+1$. Quadratic equations graph as parabolas, linear equations as lines.
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What is the meaning of the equation $y=mx+b$ in graphing terms?
What is the meaning of the equation $y=mx+b$ in graphing terms?
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Its solutions form a line with slope $m$ and $y$-intercept $b$. Slope-intercept form directly shows the line's characteristics.
Its solutions form a line with slope $m$ and $y$-intercept $b$. Slope-intercept form directly shows the line's characteristics.
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What is the solution set description for the equation $x^2+y^2=9$?
What is the solution set description for the equation $x^2+y^2=9$?
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All points $(x,y)$ with distance $3$ from the origin. This describes a circle with radius $3$ centered at origin.
All points $(x,y)$ with distance $3$ from the origin. This describes a circle with radius $3$ centered at origin.
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What is the solution set description for the equation $x+y=6$?
What is the solution set description for the equation $x+y=6$?
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All points $(x,y)$ whose coordinates sum to $6$. The coordinates must satisfy the addition constraint.
All points $(x,y)$ whose coordinates sum to $6$. The coordinates must satisfy the addition constraint.
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What is the solution set description for the equation $y=2x+3$?
What is the solution set description for the equation $y=2x+3$?
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All points $(x,y)$ with $y$ equal to $2x+3$. Each $x$-value produces exactly one corresponding $y$-value.
All points $(x,y)$ with $y$ equal to $2x+3$. Each $x$-value produces exactly one corresponding $y$-value.
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Identify the graph type of $y=-1$ as a set of solutions in the plane.
Identify the graph type of $y=-1$ as a set of solutions in the plane.
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A horizontal line consisting of all points $(x,-1)$. Horizontal lines contain all points with the same $y$-coordinate.
A horizontal line consisting of all points $(x,-1)$. Horizontal lines contain all points with the same $y$-coordinate.
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Identify the graph type of $x=2$ as a set of solutions in the plane.
Identify the graph type of $x=2$ as a set of solutions in the plane.
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A vertical line consisting of all points $(2,y)$. Vertical lines contain all points with the same $x$-coordinate.
A vertical line consisting of all points $(2,y)$. Vertical lines contain all points with the same $x$-coordinate.
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Which ordered pair is on the graph of $y=-3$ : $(0,-3)$ or $(-3,0)$?
Which ordered pair is on the graph of $y=-3$ : $(0,-3)$ or $(-3,0)$?
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$(0,-3)$. Horizontal line equations have form $y = k$.
$(0,-3)$. Horizontal line equations have form $y = k$.
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What is the meaning of the statement "the graph is a curve" for an equation?
What is the meaning of the statement "the graph is a curve" for an equation?
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The solution set forms a non-straight path of points. Non-linear equations create curved solution paths.
The solution set forms a non-straight path of points. Non-linear equations create curved solution paths.
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Identify whether $(2,5)$ is on the graph of $y=2x+1$.
Identify whether $(2,5)$ is on the graph of $y=2x+1$.
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Yes, because $5=2(2)+1$. Substitution confirms: $5 = 2(2) + 1 = 5$.
Yes, because $5=2(2)+1$. Substitution confirms: $5 = 2(2) + 1 = 5$.
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Identify whether $(2,4)$ is on the graph of $y=2x+1$.
Identify whether $(2,4)$ is on the graph of $y=2x+1$.
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No, because $4\ne 2(2)+1$. Substitution shows: $4 \neq 2(2) + 1 = 5$.
No, because $4\ne 2(2)+1$. Substitution shows: $4 \neq 2(2) + 1 = 5$.
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Identify whether $(-1,3)$ is on the graph of $y=-2x+1$.
Identify whether $(-1,3)$ is on the graph of $y=-2x+1$.
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Yes, because $3=-2(-1)+1$. Substitution confirms: $3 = -2(-1) + 1 = 3$.
Yes, because $3=-2(-1)+1$. Substitution confirms: $3 = -2(-1) + 1 = 3$.
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Identify whether $(0,-4)$ is on the graph of $y=x-4$.
Identify whether $(0,-4)$ is on the graph of $y=x-4$.
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Yes, because $-4=0-4$. Substitution confirms: $-4 = 0 - 4 = -4$.
Yes, because $-4=0-4$. Substitution confirms: $-4 = 0 - 4 = -4$.
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Identify whether $(3,0)$ is on the graph of $x+y=3$.
Identify whether $(3,0)$ is on the graph of $x+y=3$.
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Yes, because $3+0=3$. Substitution confirms: $3 + 0 = 3$.
Yes, because $3+0=3$. Substitution confirms: $3 + 0 = 3$.
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What is the solution set description for the equation $y=x^2-1$?
What is the solution set description for the equation $y=x^2-1$?
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All points $(x,y)$ with $y$ equal to $x^2-1$. Each $x$-value produces a corresponding squared value minus one.
All points $(x,y)$ with $y$ equal to $x^2-1$. Each $x$-value produces a corresponding squared value minus one.
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What is the quickest check to decide if $(x,y)$ is on the graph of an equation?
What is the quickest check to decide if $(x,y)$ is on the graph of an equation?
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Substitute $x$ and $y$ and see if the equation is true. Direct substitution determines membership in the solution set.
Substitute $x$ and $y$ and see if the equation is true. Direct substitution determines membership in the solution set.
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What is the coordinate plane used for when graphing equations in two variables?
What is the coordinate plane used for when graphing equations in two variables?
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To display all solution points $(x,y)$ as a geometric set. It provides a visual representation of solution relationships.
To display all solution points $(x,y)$ as a geometric set. It provides a visual representation of solution relationships.
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What does it mean to "plot" a solution of an equation in two variables?
What does it mean to "plot" a solution of an equation in two variables?
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Place the point $(x,y)$ on the coordinate plane. Mark the solution's location on the coordinate system.
Place the point $(x,y)$ on the coordinate plane. Mark the solution's location on the coordinate system.
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What does the equation $y=k$ (constant) graph as?
What does the equation $y=k$ (constant) graph as?
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A horizontal line at $y=k$. All points have the same $y$-coordinate value.
A horizontal line at $y=k$. All points have the same $y$-coordinate value.
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