Question

(a) find the intercepts of the graph of each equation and (b) graph the equation. $$ -0.3 x+0.4 y=1.2 $$

Short answer

The y-intercept is (0, 3) and the x-intercept is (-4, 0).

Step-by-step solution

- Find the y-intercept

To find the y-intercept, set \(x = 0\) and solve for \(y\). Starting with the equation: \[-0.3x + 0.4y = 1.2\]Substitute \(x = 0\):\[-0.3(0) + 0.4y = 1.2\]Simplify to find \(y\):\[0.4y = 1.2\]\[y = \frac{1.2}{0.4} = 3\]So, the y-intercept is \( (0, 3) \).

- Find the x-intercept

To find the x-intercept, set \(y = 0\) and solve for \(x\). Starting with the equation:\[-0.3x + 0.4y = 1.2\]Substitute \(y = 0\):\[-0.3x + 0.4(0) = 1.2\]Simplify to find \(x\):\[-0.3x = 1.2\]\[x = \frac{1.2}{-0.3} = -4\]So, the x-intercept is \( (-4, 0) \).

- Graph the equation

To graph the equation, plot the intercepts found in the previous steps: \( (0, 3) \) and \( (-4, 0) \). Draw a straight line passing through these points. This line represents the equation \(-0.3x + 0.4y = 1.2\).

Key concepts

Finding Intercepts

Finding intercepts is a crucial skill when working with linear equations. Intercepts are the points where the graph crosses the x-axis and y-axis.
To find the **y-intercept**, you set the value of x to 0 and solve for y. In this case, starting with the equation \[ -0.3x + 0.4y = 1.2 \], and substituting x = 0, you get \[ 0.4y = 1.2 \] which simplifies to y = 3. So, the y-intercept is (0, 3).
For the **x-intercept**, set y to 0 and solve for x. Using the same equation \[ -0.3x + 0.4y = 1.2 \], and substituting y = 0, you get \[ -0.3x = 1.2 \], which simplifies to x = -4. Therefore, the x-intercept is (-4, 0).
By finding these intercepts, we get two key points to help plot the graph of the linear equation.

Plotting Points

Plotting points is an essential step when graphing linear equations. Start with the intercepts you found in the previous step. These points are (0, 3) for the y-intercept and (-4, 0) for the x-intercept.
To plot the points:

• Locate (0, 3) on the graph. This point is 3 units up from the origin on the y-axis.
• Next, find (-4, 0). This point is 4 units to the left of the origin on the x-axis.

By accurately placing these points on the coordinate plane, you create a visual representation that can help you understand the behavior of the linear equation.
Once the points are plotted, draw a straight line through them. The line represents all the solutions to the equation \[ -0.3x + 0.4y = 1.2 \].

Solving Linear Equations

Solving linear equations is about finding the values of x and y that make the equation true. For this exercise, the steps involve finding intercepts and then plotting the equation.
Start by rearranging the equation if necessary to isolate variables and simplify solving. With our example \[ -0.3x + 0.4y = 1.2 \], we didn't need further rearranging, but generally, make sure your steps lead to isolating x, y, or constants easily.
Finding intercepts by setting x or y to 0 is often the first step. This makes it simpler to compute because one variable drops out, making the equation easier to solve.
After finding the intercepts, confirm by substituting these points back into the original equation. If both checks are correct, you've solved the equation correctly.
Ultimately, plotting these solutions on a graph gives you a visual tool to understand the entire set of solutions for that equation. This combination of algebraic and graphical methods can strengthen your grasp of linear equations.