$y=-\frac{3}{2} x+2$
Answer (1)
The equation of the line is y = − 2 3 x + 2 .
The slope of the line is − 2 3 and the y-intercept is 2.
The equation in standard form is 3 x + 2 y = 4 .
The x-intercept is 3 4 .
The key characteristics of the line are: y = − 2 3 x + 2 .
Explanation
Analyzing the Equation We are given the equation of a line: y = − 2 3 x + 2 . Our goal is to analyze this equation and extract key information about the line.
Identifying Slope and Y-Intercept First, let's identify the slope and y-intercept of the line. The equation is in slope-intercept form, which is y = m x + b , where m is the slope and b is the y-intercept. In our case, m = − 2 3 and b = 2 . So, the slope of the line is − 2 3 and the y-intercept is 2.
Converting to Standard Form Next, let's express the equation in the standard form A x + B y = C . To do this, we can multiply both sides of the equation by 2 to eliminate the fraction: 2 y = − 3 x + 4 . Then, we can add 3 x to both sides to get 3 x + 2 y = 4 . So, the equation in standard form is 3 x + 2 y = 4 .
Finding the X-Intercept Now, let's find the x-intercept by setting y = 0 and solving for x . Substituting y = 0 into the equation 3 x + 2 y = 4 , we get 3 x + 2 ( 0 ) = 4 , which simplifies to 3 x = 4 . Dividing both sides by 3, we find x = 3 4 . So, the x-intercept is 3 4 .
Summary of Findings In summary, the line has a slope of − 2 3 , a y-intercept of 2, and an x-intercept of 3 4 . The equation in standard form is 3 x + 2 y = 4 .
Examples
Understanding linear equations is crucial in many real-world applications. For instance, if you are tracking the depreciation of an asset over time, a linear equation can model the decreasing value. If a car's value decreases by $3 , 000 each year from an initial value of $20 , 000 , the equation y = − 3000 x + 20000 models its value y after x years. Analyzing the slope and intercepts helps predict the car's value at any point in time and when it will become worthless.
