QuestionDetermine if these equations express $y$ as a function of $x$ : $x^{2}+y^{2}=25$ , $2 x+3 y=10$ , $x=y^{2}-25$ .

Studdy Solution

STEP 1

Assumptions1. A function is a relation in which each input (x-value) has exactly one output (y-value). . To determine whether a relationship represents y as a function of x, we can use the vertical line test. If any vertical line intersects the graph of the relation at more than one point, then the relation is not a function.

STEP 2

For the first equation, we have $x^{2}+y^{2}=25$ This equation represents a circle with radius5 centered at the origin.

STEP 3

If we draw a vertical line anywhere on the graph of this equation, it will intersect the graph at two points. Therefore, the first equation does not represent y as a function of x.

STEP 4

For the second equation, we have $2x+3y=10$ This equation represents a line in the xy-plane.

STEP 5

If we draw a vertical line anywhere on the graph of this equation, it will intersect the graph at exactly one point. Therefore, the second equation does represent y as a function of x.

STEP 6

For the third equation, we have $x=y^{2}-25$ This equation represents a parabola that opens to the right if y is the independent variable.

STEP 7

If we draw a vertical line anywhere on the graph of this equation, it will intersect the graph at more than one point. Therefore, the third equation does not represent y as a function of x.
So, only the second equation represents y as a function of x.

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QuestionDetermine if these equations express $y$ as a function of $x$ : $x^{2}+y^{2}=25$ , $2 x+3 y=10$ , $x=y^{2}-25$ .

Studdy Solution

STEP 1

STEP 2

For the first equation, we have $x^{2}+y^{2}=25$ This equation represents a circle with radius5 centered at the origin.

STEP 3

If we draw a vertical line anywhere on the graph of this equation, it will intersect the graph at two points. Therefore, the first equation does not represent y as a function of x.

STEP 4

For the second equation, we have $2x+3y=10$ This equation represents a line in the xy-plane.

STEP 5

If we draw a vertical line anywhere on the graph of this equation, it will intersect the graph at exactly one point. Therefore, the second equation does represent y as a function of x.

STEP 6

For the third equation, we have $x=y^{2}-25$ This equation represents a parabola that opens to the right if y is the independent variable.

STEP 7

If we draw a vertical line anywhere on the graph of this equation, it will intersect the graph at more than one point. Therefore, the third equation does not represent y as a function of x.
So, only the second equation represents y as a function of x.

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QuestionDetermine if these equations express yyy as a function of xxx: x2+y2=25x^{2}+y^{2}=25x2+y2=25, 2x+3y=102 x+3 y=102x+3y=10, x=y2−25x=y^{2}-25x=y2−25.

Studdy Solution

STEP 1

STEP 2

For the first equation, we havex2+y2=25x^{2}+y^{2}=25x2+y2=25This equation represents a circle with radius5 centered at the origin.

STEP 3

If we draw a vertical line anywhere on the graph of this equation, it will intersect the graph at two points. Therefore, the first equation does not represent y as a function of x.

STEP 4

For the second equation, we have2x+3y=102x+3y=102x+3y=10This equation represents a line in the xy-plane.

STEP 5

If we draw a vertical line anywhere on the graph of this equation, it will intersect the graph at exactly one point. Therefore, the second equation does represent y as a function of x.

STEP 6

For the third equation, we havex=y2−25x=y^{2}-25x=y2−25This equation represents a parabola that opens to the right if y is the independent variable.

STEP 7

If we draw a vertical line anywhere on the graph of this equation, it will intersect the graph at more than one point. Therefore, the third equation does not represent y as a function of x.So, only the second equation represents y as a function of x.

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QuestionDetermine if these equations express $y$ as a function of $x$ : $x^{2}+y^{2}=25$ , $2 x+3 y=10$ , $x=y^{2}-25$ .

For the first equation, we have $x^{2}+y^{2}=25$ This equation represents a circle with radius5 centered at the origin.

For the second equation, we have $2x+3y=10$ This equation represents a line in the xy-plane.

For the third equation, we have $x=y^{2}-25$ This equation represents a parabola that opens to the right if y is the independent variable.

If we draw a vertical line anywhere on the graph of this equation, it will intersect the graph at more than one point. Therefore, the third equation does not represent y as a function of x.
So, only the second equation represents y as a function of x.