What is the $y$-intercept for the equation $5x - 2y = -4$?
Answer (1)
To find the y -intercept, set x = 0 in the equation 5 x − 2 y = − 4 .
Substitute x = 0 : 5 ( 0 ) − 2 y = − 4 , which simplifies to − 2 y = − 4 .
Solve for y : y = − 2 − 4 = 2 .
The y -intercept is ( 0 , 2 ) .
Explanation
Understanding the Problem We are given the equation 5 x − 2 y = − 4 and asked to find the y -intercept. The y -intercept is the point where the line crosses the y -axis. This occurs when x = 0 .
Substituting x=0 To find the y -intercept, we substitute x = 0 into the equation 5 x − 2 y = − 4 . This gives us 5 ( 0 ) − 2 y = − 4 .
Simplifying the Equation Simplifying the equation, we have 0 − 2 y = − 4 , which is the same as − 2 y = − 4 .
Solving for y To solve for y , we divide both sides of the equation by − 2 : − 2 − 2 y = − 2 − 4 This simplifies to y = 2 .
Finding the y-intercept Therefore, the y -intercept is the point ( 0 , 2 ) .
Examples
Understanding y-intercepts is crucial in many real-world applications. For instance, imagine you're tracking the water level in a tank over time. The y-intercept represents the initial water level when you started measuring (time = 0). Similarly, in business, if you're analyzing a cost function, the y-intercept could represent the fixed costs that your company incurs regardless of production volume. Identifying these initial values helps in making informed decisions and predictions.
