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Chapter 1: Problem 72

Use intercepts to graph equation. $$6 x-3 y+15=0$$

Short Answer

Expert verified
The x-intercept is \(-2.5\) and the y-intercept is \(5\). Therefore, the graph of the equation will be a straight line crossing the x-axis at \(-2.5\) and the y-axis at \(5\).

Step by step solution

01

Find the x-intercept

Set \(y = 0\) in the equation: \(6x - 3*0 + 15 = 0\). Solving for \(x\) gives \(x=-15/6=-2.5\). Thus, the x-intercept is \(-2.5\).
02

Find the y-intercept

Set \(x = 0\) in the equation: \(6*0 - 3y + 15 = 0\). Solving for \(y\) gives \(y=-15/-3=5\). Thus, the y-intercept is \(5\).
03

Graph the Equation

Plot the two points (\(-2.5, 0\)) and \((0, 5)\) on the graph. These points represent the x-intercept and y-intercept respectively. Then, draw a line through both points 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
When we talk about the x-intercept of a linear equation, we're referring to the point at which the line crosses the x-axis. In algebra, the x-intercept is found by setting the variable y equal to zero and solving for x. This is because any point on the x-axis will have a y-coordinate of zero.

For instance, in the given exercise with equation 6x - 3y + 15 = 0, we get the x-intercept by solving the equation 6x + 15 = 0, which simplifies to x = -2.5. Thus, the x-intercept is the point (-2.5, 0). It represents where the line touches the x-axis, and it's a crucial point for graphing the entire line.
Y-Intercept
Similarly, the y-intercept is the point where the line crosses the y-axis. To find the y-intercept, we set the variable x to zero and solve for y. Any point on the y-axis has an x-coordinate of zero, which is why this approach works.

Looking at our equation 6x - 3y + 15 = 0, when x is zero, the equation becomes -3y + 15 = 0. Solving gives us y = 5, meaning the y-intercept is at (0, 5). It's the point where the line will cross the y-axis and is equally important in plotting the graph of the equation as the x-intercept.
Plotting Intercepts
Once the intercepts are found, plotting intercepts on the Cartesian plane is the next step. Start by drawing two perpendicular lines representing the x-axis (horizontal) and y-axis (vertical). Mark the x-intercept by placing a point at x = -2.5 on the x-axis. Then, mark the y-intercept by placing a point at y = 5 on the y-axis.

These intercepts work as guides to easily draw the line representing the linear equation. By plotting both the x-intercept and the y-intercept, we have two points which define a straight line. After marking these two points, a ruler can then be used to draw a line that passes through both, giving a graphical representation of the equation.
Solving Linear Equations
The process of solving linear equations involves finding values for the variables that make the equation true. Linear equations are of the form ax + by + c = 0, where a, b, and c are constants. To solve them, we often use techniques such as substitution, elimination, or graphing.

A linear equation in two variables, like the one in our exercise, represents a straight line when graphed. Graphing helps to visualize the solutions and can also be useful in finding the intercepts. To draw a line, you only need two points, which is why identifying the x-intercept and y-intercept is so effective. Once those points are established, you have everything you need to graph the equation and visualize all its possible solutions.

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Most popular questions from this chapter

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