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Chapter 3: Problem 19

Find the \(x\) - and \(y\) -intercepts for the graph of each equation. $$ 5 x-2 y=20 $$

Short Answer

Expert verified
The x-intercept is 4 and the y-intercept is -10.

Step by step solution

01

Find the x-intercept

To find the x-intercept, set y = 0 in the equation and solve for x. Start with the equation: \rm{eq}=5x - 2y = 20 f{eq}\text{x -intercept }y=0}status eq eq simplified:{\text{x -intercept +solution eq eq simplified:{\text{eq balance needs solving:}} eq -2)(0):Now, substitute y = 0 into the equation and simplify:\[5x - 2(0) = 20\]\[5x = 20\]Next, solve for x by dividing both sides by 5:\[x = \frac{20}{5} = 4\]So, the x-intercept is 4.
02

Find the y-intercept

To find the y-intercept, set x = 0 in the equation and solve for y. Start with the equation:\[5x - 2y = 20\]Now, substitute x = 0 into the equation and simplify:\[5(0) - 2y = 20\]\[-2y = 20\]Next, solve for y by dividing both sides by -2:\[y = \frac{20}{-2} = -10\]So, the y-intercept is -10.

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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 a point where the graph of an equation crosses the x-axis. At this point, the value of y is always zero. To find the x-intercept for the equation \(5x - 2y = 20\), follow these steps:

  • Set y = 0 in the equation.
  • Substitute y = 0 into the equation: \(5x - 2(0) = 20\).
  • Simplify the equation: \(5x = 20\).
  • Solve for x by dividing both sides by 5: \(x = \frac{20}{5} = 4\).
So, the x-intercept is at (4, 0). This means that the graph of the equation crosses the x-axis at the point (4, 0). Understanding how to find the x-intercept helps in graphing linear equations accurately.
y-intercept
The **y-intercept** is a point where the graph of an equation crosses the y-axis. At this point, the value of x is always zero. To find the y-intercept for the equation \(5x - 2y = 20\), follow these steps:

  • Set x = 0 in the equation.
  • Substitute x = 0 into the equation: \(5(0) - 2y = 20\).
  • Simplify the equation: \(-2y = 20\).
  • Solve for y by dividing both sides by -2: \(y = \frac{20}{-2} = -10\).
So, the y-intercept is at (0, -10). This means that the graph of the equation crosses the y-axis at the point (0, -10). Knowing how to find the y-intercept is essential for graphing linear equations effectively.
linear equations
Linear equations represent straight lines on a graph, and they can be written in different forms. One common form is the standard form, \(Ax + By = C\). Here are some key points about linear equations:

  • A linear equation creates a straight line when graphed.
  • The equation \(5x - 2y = 20\) is in standard form, where A = 5, B = -2, and C = 20.
  • To graph a linear equation, it is often helpful to find the x- and y-intercepts, as we did earlier.
  • These intercepts give you two points that the line will pass through, making it easier to draw the line accurately.
Linear equations are fundamental in algebra and are used to describe relationships between variables that change in a linear fashion. Being able to find intercepts and understand these equations helps in solving and graphing more complex mathematical problems.

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Most popular questions from this chapter

\(3 x-5 y=-1\) \(5 x+3 y=2\)

Describe what the graph of each linear equation will look like in the coordinate plane. (Hint: Rewrite the equation if necessary so that it is in a more recognizable form.) $$ x-10=1 $$

Graph each linear equation. $$ y=-1 $$

Find the \(x\) - and \(y\) -intercepts for the graph of each equation. $$ 3 x+y=0 $$

Solve each problem. Suppose that it costs a flat fee of \(\$ 20\) plus \(\$ 15\) per day to rent a pressure washer. Therefore, the cost \(y\) in dollars to rent the pressure washer for \(x\) days is given by the linear equation $$ y=15 x+20 $$ Express each of the following as an ordered pair. (a) When the washer is rented for 5 days, the cost is \(\$ 95\). (b) When the cost is \(\$ 110\), the washer is rented for 6 days.
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