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Chapter 7: Problem 5

Use the \(x\) - and \(y\)-intercepts to graph each linear equation. \(2 x+y=6\)

Short Answer

Expert verified
The graph of the equation 2x+y=6 will pass through points (3,0) and (0,6)

Step by step solution

01

Find the x-intercept

To find the x-intercept, you need to set y=0 in the initial equation. So, by substituting y=0 into 2x+y=6, the equation simplifies to 2x=6, which further simplifies to x=3 when solved for x.
02

Find the y-intercept

To find the y-intercept, it is required to set x=0 in the initial equation. So, by substituting x=0 into 2x+y=6, the equation simplifies to y=6, indicating that the y-intercept is 6.
03

Plot the intercepts

Plot the x-intercept at (3,0) and the y-intercept at (0,6) on the coordinate plane. Draw a line that passes through these two points.
04

Verify the graph

Check to make sure that the graph is a straight line and goes through both intercept points. This will determine if the line effectively represents the initial equation.

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Key Concepts

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

x-intercept
Understanding intercepts is essential in graphing linear equations. The \(x\)-intercept is a key feature of linear equations, representing the point where the line crosses the x-axis on a graph. To find this point, you set \(y = 0\) in the equation. Simplifying the equation \(2x + y = 6\) with \(y\) set to zero, you get \(2x = 6\). Solve for \(x\) by dividing both sides by 2, which gives you \(x = 3\). Thus, the \(x\)-intercept is at the coordinate \((3, 0)\).Key points to remember:
  • The \(x\)-intercept occurs where \(y = 0\).
  • Its coordinate will always have a zero in the y-position, such as \((3, 0)\).
  • It represents the line's crossing point over the x-axis.
y-intercept
The \(y\)-intercept is another crucial element for graphing linear equations. It characterizes the point where the line crosses the y-axis. To determine the \(y\)-intercept, you set \(x = 0\) in the equation. Looking at the equation \(2x + y = 6\), when \(x = 0\), it simplifies to \(y = 6\). This tells you that the \(y\)-intercept is located at \((0, 6)\).Important aspects of the \(y\)-intercept include:
  • The \(y\)-intercept occurs where \(x = 0\).
  • Its coordinate will consistently have a zero in the x-position, such as \((0, 6)\).
  • This point signifies where the line cuts across the y-axis.
coordinate plane
The coordinate plane is a fundamental platform for graphing linear equations. It's a two-dimensional space formed by two perpendicular lines: the x-axis, which runs horizontally, and the y-axis, which runs vertically. These axes divide the plane into four quadrants. The origin, indicated by the point \((0, 0)\), is where these axes intersect.When plotting points of intercepts and lines:
  • Each point on the coordinate plane is designated by an \((x, y)\) pair.
  • The \(x\)-coordinate expresses the horizontal position while the \(y\)-coordinate reflects the vertical position.
  • To graph a linear equation, like \(2x + y = 6\), you identify and plot these intercepts on the plane to visualize the line.
linear equation
A linear equation is a mathematical expression that establishes a straight line when graphed on a coordinate plane. These equations can typically be written in the form \(y = mx + b\), where \(m\) is the slope of the line and \(b\) is the y-intercept.Examining the equation \(2x + y = 6\):
  • This fits the standard form of a linear equation \(Ax + By = C\).
  • We graph it by determining its x-intercept and y-intercept.
  • Once plotted, you should see a continuous straight line through these points.
Linear equations are foundational in algebra, helping to describe relationships between variables, and are instrumental in both basic and complex mathematical modeling.

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Most popular questions from this chapter

In Exercises 27-28, use the directions for Exercises 9-14 to graph each quadratic function. Use the quadratic formula to find \(x\)-intercepts, rounded to the nearest tenth. \(f(x)=-2 x^{2}+4 x+5\)

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If \(x\) represents height, in inches, and y represents weight, in pounds, the healthy weight region can be modeled by the following system of linear inequalities: $$ \left\\{\begin{array}{l} 5.3 x-y \geq 180 \\ 4.1 x-y \leq 140 \end{array}\right. $$ Use this information to solve Exercises 45-48. Is a person in this age group who is 6 feet tall weighing 205 pounds within the healthy weight region?

A television manufacturer makes rear-projection and plasma televisions. The profit per unit is \(\$ 125\) for the rearprojection televisions and \(\$ 200\) for the plasma televisions. a. Let \(x=\) the number of rear-projection televisions manufactured in a month and \(y=\) the number of plasma televisions manufactured in a month. Write the objective function that describes the total monthly profit. b. The manufacturer is bound by the following constraints: \- Equipment in the factory allows for making at most 450 rear-projection televisions in one month. \- Equipment in the factory allows for making at most 200 plasma televisions in one month. \- The cost to the manufacturer per unit is \(\$ 600\) for the rear-projection televisions and \(\$ 900\) for the plasma televisions. Total monthly costs cannot exceed \(\$ 360,000 .\) Write a system of three inequalities that describes these constraints. c. Graph the system of inequalities in part (b). Use only the first quadrant and its boundary, because \(x\) and \(y\) must both be nonnegative. d. Evaluate the objective function for total monthly profit at each of the five vertices of the graphed region. [The vertices should occur at \((0,0),(0,200),(300,200)\), \((450,100)\), and \((450,0)\).] e. Complete the missing portions of this statement: The television manufacturer will make the greatest profit by manufacturing rear-projection televisions each month and plasma televisions each month. The maximum monthly profit is $\$$

Graph the solution set of each system of inequalities. \(\left\\{\begin{array}{l}x \geq 4 \\ y \leq 2\end{array}\right.\)
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