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Chapter 2: Problem 104

(a) find the intercepts of the graph of each equation and (b) graph the equation. $$ 5 x+3 y=18 $$

Short Answer

Expert verified
(3.6, 0) and (0, 6); graph these intercepts and draw the line.

Step by step solution

01

- Find the x-intercept

To find the x-intercept, set \(y = 0\) and solve for \(x\).\[5x + 3(0) = 18\]\[5x = 18\]\[x = \frac{18}{5}\]\[x = 3.6\]So, the x-intercept is \((3.6, 0)\).
02

- Find the y-intercept

To find the y-intercept, set \(x = 0\) and solve for \(y\).\[5(0) + 3y = 18\]\[3y = 18\]\[y = \frac{18}{3}\]\[y = 6\]So, the y-intercept is \((0, 6)\).
03

- Plot the intercepts

Plot the points \((3.6, 0)\) and \((0, 6)\) on a graph.
04

- Draw the line

Draw a straight line through the points \((3.6, 0)\) and \((0, 6)\) to graph the equation.

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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 a linear equation is where the graph crosses the x-axis. To find the x-intercept, set \(y = 0\) in the equation and solve for \(x\). In our exercise, we started with the equation \(5x + 3y = 18\). Setting \(y\) to \(0\), we get \(5x + 3(0) = 18\). This simplifies to \(5x = 18\), and solving for \(x\) gives \(x = \frac{18}{5}\), which is \(3.6\). Thus, the x-intercept is at the point \((3.6, 0)\). It's important to remember: x-intercepts always have the form \((x, 0)\).
y-intercept
The y-intercept of a linear equation is where the graph crosses the y-axis. To find the y-intercept, set \(x = 0\) in the equation and solve for \(y\). For our equation \(5x + 3y = 18\), setting \(x\) to \(0\), we get \(5(0) + 3y = 18\). This simplifies to \(3y = 18\), and solving for \(y\) gives \(y = 6\). Therefore, the y-intercept is at the point \((0, 6)\). As with x-intercepts, y-intercepts always have the form \((0, y)\).
graphing linear equations
Once you find the intercepts, graphing a linear equation becomes simpler. Start by plotting the intercepts on the coordinate plane. For our example, plot \((3.6, 0)\) (the x-intercept) and \((0, 6)\) (the y-intercept).
  • First, draw a dot at \((3.6, 0)\) on the x-axis.
  • Next, draw another dot at \((0, 6)\) on the y-axis.
Finally, draw a straight line through both points. This line represents the graph of our equation \(5x + 3y = 18\). Remember, the line extends infinitely in both directions.
solving equations
Solving linear equations involves finding the values of \(x\) and \(y\) that make the equation true. Common steps include:
  • Isolate one variable by setting either \(x\) or \(y\) to \(0\), which helps find the intercepts.
  • Use basic algebraic operations like addition, subtraction, multiplication, or division to solve for the other variable.
In our example, to find the x-intercept, we set \(y = 0\) and solved for \(x\): \(5x = 18 鈫� x = 3.6\). To find the y-intercept, we set \(x = 0\) and solved for \(y\): \(3y = 18 鈫� y = 6\). These methods help solve and understand any linear equation better.

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