Question

(a) find the intercepts of the graph of each equation and (b) graph the equation. $$ 7 x+2 y=21 $$

Short answer

Intercepts: (3, 0) and (0, 10.5)

Step-by-step solution

- Find the x-intercept

To find the x-intercept, set y to 0 and solve for x. Start with the equation: \[ 7x + 2y = 21 \]Set \( y = 0 \): \[ 7x + 2(0) = 21 \]Simplify: \[ 7x = 21 \]Divide both sides by 7: \[ x = 3 \]So, the x-intercept is (3, 0).

- Find the y-intercept

To find the y-intercept, set x to 0 and solve for y. Start with the equation: \[ 7x + 2y = 21 \]Set \( x = 0 \): \[ 7(0) + 2y = 21 \]Simplify: \[ 2y = 21 \]Divide both sides by 2: \[ y = 10.5 \]So, the y-intercept is (0, 10.5).

- Plot the intercepts on the graph

Plot the points (3, 0) and (0, 10.5) on the coordinate plane.

- Draw the line

Using the two intercepts, draw a straight line through both points. This line represents the equation \( 7x + 2y = 21 \).

Key concepts

intercepts

Intercepts are crucial points on a graph where a line crosses the axes. They help us understand the behavior of an equation easily. The two main intercepts are:

• The x-intercept, where the line crosses the x-axis.
• The y-intercept, where the line crosses the y-axis.

To find these intercepts for a given equation, we follow straightforward steps. For the x-intercept, set y to 0 and solve for x. For the y-intercept, set x to 0 and solve for y. Let's consider the equation: \[ 7x + 2y = 21 \] To find the x-intercept, set \['y' = 0\]: \[ 7x + 2(0) = 21 \]This simplifies to: \[ 7x = 21 \]Divide both sides by 7: \[ x = 3 \]So, the x-intercept is at (3, 0). For the y-intercept, set \['x' = 0\]: \[ 7(0) + 2y = 21 \]Simplify to: \[ 2y = 21 \]Divide both sides by 2: \[ y = 10.5 \]Thus, the y-intercept is at (0, 10.5). Properly finding and plotting these intercepts provides the framework for graphing and understanding the behavior of equations.

graphing equations

Graphing equations is an excellent visual method to interpret linear equations. It helps convert the abstract mathematical concepts into clear, visual insights. Here's how you do it:

• Step 1: Identify the intercepts - the points where the graph crosses the axes. For our example, the intercepts are (3, 0) and (0, 10.5).
• Step 2: Plot these intercepts on the coordinate plane. Place point (3,0) on the x-axis and point (0,10.5) on the y-axis.
• Step 3: Draw a straight line through the plotted points. This line represents the equation.

By using the intercepts (3,0) and (0,10.5), you can correctly draw the graph of the equation. Graphing allows us to quickly visualize and understand the relationship between variables in the equation.
Additionally, the slope and intercepts can provide many insights, such as predicting values and understanding trends. The equation \[ 7x + 2y = 21 \] is effectively visualized on a graph by connecting these intercept points.

solving linear equations

Solving linear equations involves finding the values of variables that satisfy the equation. The general form of a linear equation is \[\text{ax} + \text{by} = \text{c}\]. Here's how to solve the given linear equation step-by-step: \[ 7x + 2y = 21 \]

• Start by solving for one variable in terms of the other. For example, solve for y when x = 0 to find the y-intercept.
• Substitute these values back into the equation to solve for the other variable when one variable is set to zero.
• Use these solutions to interpret the equation's behavior graphically.

We found that when y = 0, x = 3, providing the x-intercept (3, 0); and when x = 0, y = 10.5, providing the y-intercept (0, 10.5). These solutions allow us to plot the intercepts and draw the line representing the equation on the coordinate plane. Solving linear equations is a fundamental skill that aids in understanding more complex algebraic concepts and real-world applications.