Problem 18 - Understanding Elementary Algebra with Geometry

Question

Find the \(x\) - and \(y\) -intercepts of the equation. $$3 x=-7 y$$

Step-by-Step Solution

Answer

Both the x- and y-intercepts are at (0, 0).

Step 1: Find the y-intercept

To find the y-intercept, set x to 0 in the equation and solve for y. The equation is: \[ 3x = -7y \]Substitute x with 0:\[ 3(0) = -7y \]This simplifies to:\[ 0 = -7y \]So, solve for y:\[ y = 0 \]Thus, the y-intercept is (0, 0).

Step 2: Find the x-intercept

To find the x-intercept, set y to 0 in the equation and solve for x. The equation is:\[ 3x = -7y \]Substitute y with 0:\[ 3x = -7(0) \]This simplifies to:\[ 3x = 0 \]So, solve for x:\[ x = 0 \]Thus, the x-intercept is (0, 0).

Key Concepts

x-intercept

The x-intercept of a linear equation is the point where the graph of the equation crosses the x-axis. This occurs when the y-value is zero. To find the x-intercept, you set y to 0 in the equation and solve for x.

For our equation, 3x = -7y:
1. Substitute y with 0: 3x = -7(0).
2. Simplify the equation: 3x = 0.
3. Solve for x: x = 0.
Hence, the x-intercept is at (0, 0).

Remember, the x-intercept is always in the form of (x, 0).

y-intercept

The y-intercept of a linear equation is the point where the graph crosses the y-axis. This occurs when the x-value is zero. To find the y-intercept, you set x to 0 in the equation and solve for y.

In our given equation, 3x = -7y:
1. Substitute x with 0: 3(0) = -7y.
2. Simplify the equation: 0 = -7y.
3. Solve for y: y = 0.
Hence, the y-intercept is at (0, 0).

The y-intercept is always in the form of (0, y).

linear equations

Linear equations are equations that form a straight line when graphed. They have variables that appear only to the first power and have no products of variables together. The general form of a linear equation in two variables (x and y) is:

Ax + By = C, where A, B, and C are constants.
A linear equation can often be written in different forms like slope-intercept form (y = mx + b) or standard form (Ax + By = C).

In our case, 3x = -7y can be considered to be in standard form. The coefficients indicate how changes in one variable affect the other in a consistent, linear relationship.

Understanding linear equations is crucial since many phenomena can be modeled as linear relationships.

solving equations

Solving equations involves finding the value(s) of the variables that make the equation true. Different types of equations require different methods for solving.

To solve linear equations, we use basic algebraic principles like:
1. Isolating the variable by performing inverse operations.
2. Simplifying both sides of the equation.
3. Substituting known values to find unknowns.

For our equation 3x = -7y:
1. For the y-intercept, set x to 0, and solve for y: y = 0.
2. For the x-intercept, set y to 0, and solve for x: x = 0.

Through practice, solving equations becomes more intuitive and helps in understanding how different variables interrelate.