The GeoGebra app is designed to graph lines in the form of \( ax + by = c \). When a specific line equation is input, the app provides valuable information such as the slope, y-intercept, x-intercept, and the line equation in slope-intercept form.

For example, if a line with a slope of 4 is input, the app will indicate that for every 4 units moved to the right, there is also a movement of 1 unit up. If the y-intercept of the line is 2, the line would cross the y-axis at (0, 2), and the equation of the line in slope-intercept form would be \( y = 4x + 2 \).

Overall, the GeoGebra app serves as a powerful tool for visualizing and understanding lines. It offers both a visual representation and detailed information about the line, including aspects such as the slope, y-intercept, x-intercept, and more. It can be an invaluable resource for learning about graphing lines and comprehending their properties.


Example 1: Converting \(3x + 2y = 6\) into Slope-Intercept Form

\(3x + 2y = 6\) Start with the given equation.
\(2y = -3x + 6\) Subtract \(3x\) from both sides to isolate the term with \(y\). This helps us get closer to the slope-intercept form.
\(y = -\frac{3}{2}x + 3\) Divide both sides by 2 to solve for \(y\). This gives us the slope \(-\frac{3}{2}\) and the y-intercept 3.
\(y = -\frac{3}{2}x + 3\) The equation is now in slope-intercept form \(y = mx + b\), where \(m = -\frac{3}{2}\) is the slope, and \(b = 3\) is the y-intercept.

This step-by-step process illustrates how to convert a linear equation into slope-intercept form. Understanding the slope and y-intercept of a line is essential for graphing and analyzing linear relationships.

Example 2: Converting \(4.5x – 1.2y = 3.6\) into Slope-Intercept Form

\(4.5x – 1.2y = 3.6\) Start with the given equation.
\(-1.2y = -4.5x + 3.6\) Subtract \(4.5x\) from both sides to isolate the term with \(y\). This helps us get closer to the slope-intercept form.
\(y = \frac{4.5}{1.2}x – 3\) Divide both sides by \(-1.2\) to solve for \(y\). This gives us the slope \(\frac{4.5}{1.2}\) and the y-intercept \(-3\).
\(y = \frac{15}{4}x – 3\) The equation is now in slope-intercept form \(y = mx + b\), where \(m = \frac{15}{4}\) is the slope, and \(b = -3\) is the y-intercept.

This step-by-step process illustrates how to convert a linear equation with decimal coefficients into slope-intercept form. Understanding the slope and y-intercept of a line is essential for graphing and analyzing linear relationships.

Example 3: Converting \(\frac{3}{4}x – \frac{5}{2}y = \frac{7}{8}\) into Slope-Intercept Form

\(\frac{3}{4}x – \frac{5}{2}y = \frac{7}{8}\) Start with the given equation.
\(-\frac{5}{2}y = -\frac{3}{4}x + \frac{7}{8}\) Subtract \(\frac{3}{4}x\) from both sides to isolate the term with \(y\).
\(y = \frac{2}{5} \cdot \frac{3}{4}x – \frac{2}{5} \cdot \frac{7}{8}\) Multiply both sides by \(-\frac{2}{5}\) to cancel the coefficient of \(y\).
\(y = \frac{3}{10}x – \frac{7}{20}\) Multiply the fractions by multiplying the numerators together and the denominators together.
\(y = \frac{3}{10}x – \frac{7}{20}\) The equation is now in slope-intercept form \(y = mx + b\), where \(m = \frac{3}{10}\) is the slope, and \(b = -\frac{7}{20}\) is the y-intercept.
\(m = \frac{3}{10}\) Identify the slope of the line, which represents the rate of change of \(y\) with respect to \(x\).
\(b = -\frac{7}{20}\) Identify the y-intercept of the line, which represents the value of \(y\) when \(x = 0\).
\(y = \frac{3}{10}x – \frac{7}{20}\) Summary: We have successfully converted the equation into slope-intercept form by isolating \(y\), dividing fractions, and identifying the slope and y-intercept.

This step-by-step process illustrates how to convert a linear equation with fractional coefficients into slope-intercept form. Understanding the slope and y-intercept of a line is essential for graphing and analyzing linear relationships.

Example 4: Converting \(0.5x – \frac{3}{4}y = 1.25\) into Slope-Intercept Form

\(0.5x – \frac{3}{4}y = 1.25\) Start with the given equation.
\(-\frac{3}{4}y = -0.5x + 1.25\) Subtract \(0.5x\) from both sides to isolate the term with \(y\).
\(y = \frac{4}{3} \cdot (-0.5x + 1.25)\) Multiply both sides by \(-\frac{4}{3}\) to cancel the coefficient of \(y\).
\(y = \frac{4}{3} \cdot -0.5x + \frac{4}{3} \cdot 1.25\) Distribute the \(\frac{4}{3}\) to both terms on the right side.
\(y = -\frac{2}{3}x + \frac{5}{3}\) Multiply the fractions and decimals together to simplify.
\(y = -\frac{2}{3}x + \frac{5}{3}\) The equation is now in slope-intercept form \(y = mx + b\), where \(m = -\frac{2}{3}\) is the slope, and \(b = \frac{5}{3}\) is the y-intercept.
\(m = -\frac{2}{3}\) Identify the slope of the line, which represents the rate of change of \(y\) with respect to \(x\).
\(b = \frac{5}{3}\) Identify the y-intercept of the line, which represents the value of \(y\) when \(x = 0\).
\(y = -\frac{2}{3}x + \frac{5}{3}\) Summary: We have successfully converted the equation into slope-intercept form by isolating \(y\), working with fractions and decimals, and identifying the slope and y-intercept.

This step-by-step process illustrates how to convert a linear equation with mixed coefficients into slope-intercept form. The combination of fractions and decimals adds complexity, but the process remains the same, focusing on isolating \(y\) and simplifying the expression.