Problem 27 Use intercepts and a checkpoint ... [FREE SOLUTION]

Chapter 3: Problem 27

Use intercepts and a checkpoint to graph each equation. $$2 x-y=7$$

Short Answer

Expert verified

X-intercept is (3.5,0), Y-intercept is (0,-7) and the checkpoint is (1,-5). Draw a line passing through these three points to graph the equation.

Step by step solution

Find the x-intercept

To find the x-intercept, set $y = 0$ and solve for $x$. \[2x - 0 = 7\] thus \(x =\frac{7}{2} = 3.5\].

Find the y-intercept

To find the y-intercept, set $x = 0$ and solve for $y$. \[2*0 - y = 7\] thus $-y = 7$ and \(y = -7\].

Choose a checkpoint

Choose a point not on the line determined by the intercepts for the checkpoint. For simplicity, always try to use $x = 1$ or $x = -1$ if possible. In this case, let's choose $x = 1$. Substitute $x = 1$ into the equation and solve for $y$: \[2*1 - y = 7\] hence $y = -5\]. So the point is \( (1, -5)$.

Graph the equation

Plot the points $(3.5,0)$, $(0,-7)$, and $(1,-5)$ on the graph. Sketch a line through these points. This line represents the graph of the equation $2x - y = 7$ and should cross over the y-axis at the point $0,-7$ and over the x-axis at the point $3.5,0$. Make sure all three points are on the same line.

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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 line on a coordinate plane is where the line crosses the x-axis. At this point, the value of y is always zero. To find the x-intercept of a linear equation like $2x - y = 7$, we set $y = 0$ in the equation. Let's solve it step by step:

Substitute $y = 0$ into the equation: \[2x - 0 = 7\]
Simplify this to find $x$: \[2x = 7\]
Divide each side by 2: \[x = \frac{7}{2}\]
This results in $x = 3.5$

Once solved, you have the x-intercept which is at the point $(3.5, 0)$. This point is crucial for graphing the linear equation as it is one endpoint of the line on the x-axis.

y-intercept

The y-intercept is the point where the line crosses the y-axis, making it a significant checkpoint on any linear graph. Here, the value of x is always zero. To find the y-intercept of the equation $2x - y = 7$, follow these easy steps:

Set $x = 0$ in the equation:\[2*0 - y = 7\]
Simplify to solve for $y$: \[-y = 7\]
Divide each side by -1, \[y = -7\]

So, the y-intercept is the point $(0, -7)$ on the graph. This is another critical component in plotting the line, establishing the second axis crossing to ensure accuracy in graphing.

checkpoint

A checkpoint is an additional point on the graph used to verify the accuracy of a linear equation's representation. It should lie on the line defined by the given equation and the intercepts. To choose a suitable checkpoint, pick a simple x value, such as $x = 1$ or $x = -1$. For the equation $2x - y = 7$:

Choose $x = 1$: \[2*1 - y = 7\]
Simplify to find $y$: \[2 - y = 7\]
Subtract 2: \[-y = 5\]
Multiply by -1: \[y = -5\]

This gives the checkpoint $(1, -5)$. This point should also lie on the plotted line, ensuring it is correct and consistently representing the equation.

coordinate plane

The coordinate plane is a fundamental tool for graphing equations, allowing us to visualize mathematical relationships. It's like a map with horizontal (x-axis) and vertical (y-axis) lines, where each point is given as an ordered pair $(x, y)$. Understanding the coordinate plane includes:

Quadrants: Each section created by the axes is a quadrant.
Axis lines: These are the x-axis (horizontal) and y-axis (vertical).
Point plotting: Each point is located with x and y values in the format $(x, y)$.

The points determined by intercepts and checkpoints, like $(3.5, 0)$, $(0, -7)$, and $(1, -5)$, can be plotted on this plane to easily graph the line representing the equation.

linear graphing

Linear graphing involves plotting points to draw a straight line that represents a linear equation. For an equation like $2x - y = 7$, the process is straightforward. Here's how you can do it:

Identify intercepts: Find both x and y-intercepts.
Select a checkpoint: Choose an easy additional point to confirm the line's path.
Plot points: Carefully plot all calculated points such as $(3.5, 0)$, $(0, -7)$, and $(1, -5)$.
Draw a line: Connect these points with a straight line, extending it right through each plotted point.

This line visually represents the equation on the coordinate plane, illustrating the linear relationship described by the formula. Linear graphing simplifies the equation, making trends and relationships easier to understand and analyze.

91影视

Use intercepts and a checkpoint to graph each equation. $$2 x-y=7$$

Short Answer

Step by step solution

Find the x-intercept

Find the y-intercept

Choose a checkpoint

Graph the equation

Key Concepts

x-intercept

y-intercept

checkpoint

coordinate plane

linear graphing

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