91Ó°ÊÓ

The equation \(3 x+2 y=6\) is in standard form a. Find \(x\) when \(y\) is zero. Write your answer in the form \((x, y)\). What is the significance of this point? (a) b. Find \(y\) when \(x\) is zero. Write your answer in the form \((x, y)\). What is the significance of this point? (a) c. On graph paper, plot the points you found in \(10 \mathrm{a}\) and \(\mathrm{b}\) and draw the line through these points. (a) d. Find the slope of the line you drew in \(10 \mathrm{c}\) and write a linear equation in intercept form. e. On your calculator, graph the equation you wrote in \(10 \mathrm{~d}\). Compare this graph to the one you drew on paper. Is the intercept equation equivalent to the standard-form equation? Explain why or why not. f. Symbolically show that the equation \(3 x+2 y=6\) is equivalent to your equation from \(10 \mathrm{~d}\).

Step by step solution

01

Solve for x when y is 0

Substitute \(y = 0\) into the equation \(3x + 2y = 6\). This gives \(3x = 6\). Solve for \(x\): \(x = 2\). Thus, the point is \((2, 0)\). This point is the x-intercept of the equation.

02

Solve for y when x is 0

Substitute \(x = 0\) into the equation \(3x + 2y = 6\). This gives \(2y = 6\). Solve for \(y\): \(y = 3\). Thus, the point is \((0, 3)\). This point is the y-intercept of the equation.

03

Plot points and draw the line

On graph paper, plot the points \((2, 0)\) and \((0, 3)\). Draw a straight line through these two points. This line represents the graph of the equation \(3x + 2y = 6\).

04

Find the slope and write intercept form equation

Use the formula for slope \(m = \frac{y_2 - y_1}{x_2 - x_1}\) with points \((2, 0)\) and \((0, 3)\). The slope \(m = \frac{3 - 0}{0 - 2} = -\frac{3}{2}\). The equation in intercept form is \(y = -\frac{3}{2}x + 3\).

05

Graph the intercept form equation and compare

Graph the equation \(y = -\frac{3}{2}x + 3\) using a calculator. Compare it with the line drawn on the graph paper. Both graphs should overlap completely, indicating that the intercept equation is equivalent to the standard-form equation.

06

Show symbolic equivalence

Start with the intercept form equation \(y = -\frac{3}{2}x + 3\). Manipulate it to show equivalence to the standard form:1. Multiply every term by 2 to eliminate the fraction: \(2y = -3x + 6\).2. Rearrange to: \(3x + 2y = 6\).This confirms that the intercept form and standard form equations represent the same line.

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Key Concepts

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

Standard Form

Standard form of a linear equation is a way of expressing an equation in the form \( Ax + By = C \), where \( A, B, \) and \( C \) are constants, and \( A \) and \( B \) are not both zero. This format is useful because it makes it easy to find the intercepts of the line, which can then be used to quickly draw it on a graph. By setting \( y \) to zero, the value of \( x \) can be found (x-intercept). Similarly, by setting \( x \) to zero, the value of \( y \) can be found (y-intercept). The equation \( 3x + 2y = 6 \) is in standard form which helps simplify these calculations.

Slope-Intercept Form

The slope-intercept form of a linear equation is \( y = mx + b \), where \( m \) represents the slope and \( b \) represents the y-intercept of the line. This form is particularly useful for quickly graphing a line and understanding its behavior.

• The slope \( m \) indicates the steepness or inclination of the line. A positive slope means the line is ascending as it moves from left to right, while a negative slope means it is descending.
• The y-intercept \( b \) is the point at which the line crosses the y-axis.

For the equation \( y = -\frac{3}{2}x + 3 \), which is the intercept form equivalent of the standard form, the slope \( m \) is \(-\frac{3}{2}\), indicating the line descends with a decrease in \( x \), and the y-intercept is 3.

Graphing Lines

To graph a linear equation, you can use several methods, adding flexibility to find the most convenient approach. Here's a straightforward way:1. **Identify the Intercepts**: Locate points where the line crosses the axes by setting \( x \) and \( y \) to zero. These are your x-intercept and y-intercept.
2. **Plot the Intercepts**: On a graph paper or digital tool, mark these intercepts on the respective axes.
3. **Draw the Line**: Use a ruler or straight edge to draw a line straight through these intercepts. Since a line extends indefinitely, make sure it covers a substantial part of the graph.Graphing lines from standard form can be as simple as plotting the intercepts found and connecting them. This visualization is useful to understand relationships between variables in the equation.

Intercepts

Intercepts are where the line intersects the axes. They are critical in understanding and sketching the graph of the equation:* **X-Intercept**: This is where the line crosses the x-axis. It occurs when \( y = 0 \). Solve for \( x \) to find this intercept.* **Y-Intercept**: This is where the line crosses the y-axis. It happens when \( x = 0 \). Solve for \( y \) to determine this point.The intercepts serve as easy markers to quickly and accurately draw the line on a graph. For instance, in the equation \( 3x + 2y = 6 \), setting \( y = 0 \) gives \( x = 2 \), thus the x-intercept is \((2, 0)\). Similarly, setting \( x = 0 \) gives \( y = 3 \), so the y-intercept is \((0, 3)\). Knowing these points can greatly simplify the graphing process.