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Chapter 1: Problem 44

Find any intercepts of the graph of the given equation. Determine whether the graph of the equation possesses symmetry with respect to the \(x\) -axis, \(y\) -axis, or origin. Do not graph. \(2 x+3 y=6\)

Short Answer

Expert verified
The x-intercept is (3, 0) and the y-intercept is (0, 2). No symmetry with respect to x-axis, y-axis, or origin.

Step by step solution

01

Find the x-intercept

To find the x-intercept, set \( y = 0 \) in the equation \( 2x + 3y = 6 \). Substitute \( y = 0 \) into the equation: \( 2x + 3(0) = 6 \), which simplifies to \( 2x = 6 \). Solve for \( x \) by dividing both sides by 2, giving \( x = 3 \). Therefore, the x-intercept is \((3, 0)\).
02

Find the y-intercept

To find the y-intercept, set \( x = 0 \) in the equation \( 2x + 3y = 6 \). Substitute \( x = 0 \) into the equation: \( 2(0) + 3y = 6 \), which simplifies to \( 3y = 6 \). Solve for \( y \) by dividing both sides by 3, giving \( y = 2 \). Therefore, the y-intercept is \((0, 2)\).
03

Test for symmetry about x-axis

For symmetry about the x-axis, replace \( y \) with \( -y \) in the equation and see if it remains unchanged. Substitute \( -y \) into the equation: \( 2x + 3(-y) = 6 \) simplifies to \( 2x - 3y = 6 \), which is different from \( 2x + 3y = 6 \). This means the graph does not have symmetry with respect to the x-axis.
04

Test for symmetry about y-axis

For symmetry about the y-axis, replace \( x \) with \( -x \) in the equation and see if it remains unchanged. Substitute \( -x \) into the equation: \( 2(-x) + 3y = 6 \) simplifies to \(-2x + 3y = 6 \), which is different from \( 2x + 3y = 6 \). This means the graph does not have symmetry with respect to the y-axis.
05

Test for symmetry about origin

For symmetry about the origin, replace \( x \) with \( -x \) and \( y \) with \( -y \) in the equation and check if it remains unchanged. Substitute \( -x \) and \( -y \): \( 2(-x) + 3(-y) = 6 \) simplifies to \(-2x - 3y = 6 \), which is different from \( 2x + 3y = 6 \). Thus, the graph does not have symmetry with respect to the origin.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

x-intercept
The x-intercept of an equation is where the graph crosses the x-axis. To find it, we set the value of y to zero and solve for x. This is because any point on the x-axis has a y-value of zero. For the equation \[ 2x + 3y = 6 \],setting \( y = 0 \) lets us solve:\[ 2x + 3(0) = 6 \]\[ 2x = 6 \]\[ x = 3 \]This calculation shows that the x-intercept is at the point (3, 0). Remember:
  • X-intercepts are always written as (x, 0).
  • They represent solving the equation when y is zero.
Finding x-intercepts is a critical step in understanding where a line crosses the x-axis. It helps us understand the graph's behavior.
y-intercept
The y-intercept is where the graph intersects the y-axis. In simpler terms, it's the value of y when x is set to zero. This is because any point on the y-axis has an x-value of zero. Let's look at how you find it using the same equation:\[ 2x + 3y = 6 \]Substituting \( x = 0 \), we solve:\[ 2(0) + 3y = 6 \]\[ 3y = 6 \]\[ y = 2 \]From this, we find that the y-intercept is the point (0, 2). Keep the following in mind:
  • Y-intercepts are always represented as (0, y).
  • They indicate where the graph crosses the y-axis.
Understanding y-intercepts provides insight into the behavior of a graph at that crossing point.
symmetry in equations
Symmetry in equations refers to the balance or mirroring of a graph when it is reflected across an axis or rotated around a point. Testing for symmetry can help understand the structure of the graph without actually plotting it.To check if an equation like \(2x + 3y = 6\) is symmetric about the x-axis, we replace y with -y. If the equation remains the same, it shows symmetry about the x-axis. Here, substituting gives us:\(2x - 3y = 6\), which is different, so no symmetry across the x-axis.Testing for y-axis symmetry, we replace x with -x. We get:\(-2x + 3y = 6\),which also differs, indicating no y-axis symmetry.For symmetry about the origin, both x and y are replaced by their negatives:\(-2x - 3y = 6\),still resulting in a different equation. Thus, there's no origin symmetry.
  • Symmetry about the x-axis changes y to -y.
  • Symmetry about the y-axis changes x to -x.
  • Origin symmetry alters both to their negatives.
These checks help predict the graph's appearance without drawing it.

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Most popular questions from this chapter

In Problems \(31-34\), sketch the set of points in the \(x y\) plane whose coordinates satisfy the given inequality. \(x^{2}+y^{2} \geq 9\)

Find any intercepts of the graph of the given equation. Determine whether the graph of the equation possesses symmetry with respect to the \(x\) -axis, \(y\) -axis, or origin. Do not graph. \(y=x\left(x^{2}-3\right)\)

Determine whether the following statement is true or false. Defend your answer. Every equation of the form \(x^{2}+y^{2}+a x+b y+\) \(c=0\) is a circle.

In Problems \(27-34\), use rationalization to simplify the given expression in part (a). Then, if instructed, find the indicated limit in part (b). (a) \(\frac{\sqrt{x}-3}{x-9}\) (b) \(\lim _{x \rightarrow 9} \frac{\sqrt{x}-3}{x-9}\)

Find an equation of the circle that satisfies the given conditions. center \((-4,3),\) graph tangent to the \(y\) -axis
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