Question

In Problems \(35-38\), find the slope and \(y\) -intercept of each line. \(4 x+5 y=-20\)

Short answer

Slope: \(-\frac{4}{5}\); y-intercept: \(-4\).

Step-by-step solution

Identify the equation format

The given equation is in standard form: \(4x + 5y = -20\). To find the slope and y-intercept, we first need to convert it into slope-intercept form.

Convert to slope-intercept form

To convert the equation to slope-intercept form \(y = mx + b\), solve for \(y\). Start by isolating \(y\) on one side. Subtract \(4x\) from both sides: \(5y = -4x - 20\).

Solve for y

Divide each term by 5 to solve for \(y\):\[y = -\frac{4}{5}x - 4\]. This is the slope-intercept form of the equation.

Identify the slope and y-intercept

In the equation \(y = -\frac{4}{5}x - 4\), the slope \(m\) is \(-\frac{4}{5}\) and the y-intercept \(b\) is \(-4\).

Key concepts

Standard Form

The standard form of a linear equation is one of the most common formats in algebra. It is written as \(Ax + By = C\), where \(A\), \(B\), and \(C\) are integers, and \(A\) and \(B\) are not both zero. This form is useful because it clearly displays both variables on the left side of the equation and the constant on the right. To identify whether an equation is in standard form, check for the following characteristics:

• Both variables, \(x\) and \(y\), appear on the left.
• There are no fractions involved.
• The coefficients \(A\), \(B\), and \(C\) are integers.

In the exercise above, the equation \(4x + 5y = -20\) is in standard form, with \(A = 4\), \(B = 5\), and \(C = -20\). This format can be converted to other forms, like slope-intercept, to make it easier to analyze certain properties of the line, such as slope and intercept.

Slope-Intercept Form

Slope-intercept form is a way of writing linear equations that makes it easy to identify the slope and the y-intercept of the line. The general equation for this form is \(y = mx + b\), where \(m\) represents the slope, and \(b\) represents the y-intercept.To convert an equation from standard form to slope-intercept form, follow these easy steps:

• Isolate \(y\) on one side of the equation.
• Organize the expression to fit the \(y = mx + b\) format.

For example, converting the equation \(4x + 5y = -20\) involves solving for \(y\). By rearranging and simplifying the terms, you get \(y = -\frac{4}{5}x - 4\). Here, the slope \(m = -\frac{4}{5}\) and the y-intercept \(b = -4\) are immediately visible, making it easier to graph or analyze the line.

Linear Equations

Linear equations represent relationships between variables, forming straight lines when graphed. They are used extensively in mathematics to model real-world situations and solve problems. There are several key properties of linear equations to keep in mind:

• They're degree one, which means the highest exponent of the variables is one.
• Their graphs are straight lines, with the slope indicating the angle or steepness.
• They can be transformed between different forms (like standard and slope-intercept) for better convenience in solving problems.

Understanding these core characteristics helps in analyzing and utilizing linear equations effectively. In your example equation, both forms—standard and slope-intercept—help provide distinct insights into the equation's properties.

Mathematics Education

In mathematics education, the ability to work with different forms of linear equations is crucial. Students learn these concepts to solve problems and relate mathematical theory to practical applications. Hands-on exercises, like converting equations from standard to slope-intercept form, build critical thinking skills and deepen an understanding of linear relationships.

• Practicing these conversions helps students see the connection between algebraic representations and geometric interpretations.
• Teachers often emphasize the real-world applications of linear equations, such as predicting trends and understanding rates of change.

The exercise mentioned gives students practical experience with these important conversions, supporting their overall mathematical fluency and confidence.