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Chapter 3: Problem 16

Graph each linear equation by finding and plotting its intercepts See Examples 4 and \(5 .\) \(2 x+3 y=6\)

Short Answer

Expert verified
Plot the intercepts (3, 0) and (0, 2) and draw a line through them.

Step by step solution

01

Find the x-intercept

To find the x-intercept, set \(y = 0\) in the equation \(2x + 3y = 6\). By substituting \(y = 0\), we get \(2x + 3(0) = 6\), which simplifies to \(2x = 6\). Solving for \(x\), we divide both sides of the equation by 2, giving us \(x = 3\). So, the x-intercept is at the point \((3, 0)\).
02

Find the y-intercept

To find the y-intercept, set \(x = 0\) in the equation \(2x + 3y = 6\). By substituting \(x = 0\), we have \(2(0) + 3y = 6\), which simplifies to \(3y = 6\). Solving for \(y\), divide both sides by 3, giving \(y = 2\). Hence, the y-intercept is at the point \((0, 2)\).
03

Plot the Points on the Graph

Plot the x-intercept \((3, 0)\) and the y-intercept \((0, 2)\) on a coordinate grid. Place a point at \(x = 3\) on the x-axis and a point at \(y = 2\) on the y-axis.
04

Draw the Line

Draw a straight line through the points \((3, 0)\) and \((0, 2)\). This line represents the graph of the equation \(2x + 3y = 6\). Make sure the line extends beyond the intercepts, covering both the positive and negative sides of the axes.

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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 is the point where a line crosses the x-axis on a graph. To find this point, you need to set the \(y\) value to zero in the equation of the line. For the equation \(2x + 3y = 6\), substituting \(y = 0\) simplifies the equation to \(2x = 6\). Solve for \(x\) by dividing each side by 2, which gives \(x = 3\). Thus, the x-intercept is \((3, 0)\). This means the line touches the x-axis at the point \(x = 3\), when \(y\) is zero. This concept is crucial for graphing linear equations, as it offers a clear point through which the line crosses the axis.
Y-Intercept
The y-intercept is where a line intersects the y-axis on a graph. To locate this intersection, set \(x = 0\) in the line's equation. For the given equation \(2x + 3y = 6\), substitute \(x = 0\), simplifying to \(3y = 6\). Solving for \(y\) requires dividing each side by 3, resulting in \(y = 2\). Hence, the y-intercept is \((0, 2)\). This indicates that the line passes through the y-axis at \(y = 2\) when \(x\) is zero. Understanding y-intercepts helps in graphing lines, providing another pivotal point that shapes the line's path on the coordinate grid.
Coordinate Grid
A coordinate grid is a two-dimensional plane formed by the intersection of two perpendicular lines called axes: the x-axis (horizontal) and the y-axis (vertical). These axes divide the plane into four quadrants and allow for the precise plotting of points using ordered pairs \((x, y)\).
  • The x-axis usually represents the horizontal dimension.
  • The y-axis represents the vertical dimension.
Every point on this grid is defined by its position relative to the origin \((0, 0)\), where the x and y axes meet. Plotting points like \((3, 0)\) and \((0, 2)\) are essential steps in graphing linear equations. The grid helps visualize the relationship expressed by an equation and makes it easier to draw the corresponding line efficiently.
Equation Solving
Equation solving is the process of finding unknown variables that satisfy a mathematical statement. In the context of linear equations, it involves algebraic manipulations to find values for \(x\) and \(y\).
  • Start by identifying which variable you want to solve for.
  • Isolate the variable by performing inverse operations (such as addition, subtraction, multiplication, or division) on both sides of the equation.
  • Simplify as needed to find the solution.
For the equation \(2x + 3y = 6\), solving for \(x\) when \(y = 0\) led to \(x = 3\). Similarly, solving for \(y\) when \(x = 0\) gave \(y = 2\). Mastering equation solving is essential for mathematical proficiency and is especially useful for tasks like finding intercepts or predicting outcomes in various scenarios.

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Most popular questions from this chapter

Complete each ordered pair so that it is a solution of the given linear equation. See Example 5. $$ y=\frac{1}{4} x-3 ;(-8, \quad),(\quad, 1) $$

Minh, a psychology student, kept a record of how much time she spent studying for each of her 20 -point psychology quizzes and her score on each quiz. $$ \begin{array}{|l|c|c|c|c|c|c|c|c|} \hline \text { Hours Spent Studying } & 0.50 & 0.75 & 1.00 & 1.25 & 1.50 & 1.50 & 1.75 & 2.00 \\ \hline \text { Quiz Score } & 10 & 12 & 15 & 16 & 18 & 19 & 19 & 20 \\ \hline \end{array} $$ a. Write the data as ordered pairs of the form (hours spent studying, quiz score). b. In your own words, write the meaning of the ordered pair (1.25,16) C. Create a scatter diagram of the paired data. \(\mathrm{Be}\) sure to label the axes appropriately. d. What might Minh conclude from the scatter diagram?

Solve. See the Concept Checks in this section. In general, what points can have coordinates reversed and still have the same location?

Answer each exercise with true or false. The ordered pair \(\left(2, \frac{2}{3}\right)\) is a solution of \(2 x-3 y=6\)

Assume each exercise describes a linear relationship. Write the equations in slope-intercept form. In 2000 , crude oil field production in the United States was 2130 thousand barrels. In \(2007,\) U.S. crude oil field production dropped to 1850 thousand barrels. (Source: Energy Information Administration) a. Write two ordered pairs of the form (years after 2000, crude oil production). b. Assume the relationship between years after 2000 and crude oil production is linear over this period. Use the ordered pairs from part (a) to write an equation of the line relating years after 2000 to crude oil production. c. Use the linear equation from part (b) to estimate crude oil production in the United States in 2010 , if this trend were to continue.
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