Which graph matches the equation $y+6=\frac{3}{4}(x+4)$?
Answer (1)
Rewrite the given equation y + 6 = 4 3 ( x + 4 ) in slope-intercept form.
Simplify the equation to y = 4 3 x − 3 .
Identify the slope as 4 3 and the y-intercept as -3.
Determine the x-intercept by setting y = 0 and solving for x , which gives x = 4 .
The graph has a slope of 4 3 , a y-intercept of -3, and an x-intercept of 4.
Explanation
Understanding the Problem We are given the equation y + 6 = 4 3 ( x + 4 ) and asked to find the graph that matches this equation.
Converting to Slope-Intercept Form First, let's rewrite the equation in slope-intercept form, which is y = m x + b , where m is the slope and b is the y-intercept.
Distributing the Fraction Distribute the 4 3 on the right side of the equation: y + 6 = 4 3 x + 4 3 ( 4 ) which simplifies to y + 6 = 4 3 x + 3 .
Isolating y Now, isolate y by subtracting 6 from both sides: y = 4 3 x + 3 − 6 , which simplifies to y = 4 3 x − 3 .
Identifying Slope and y-intercept From the slope-intercept form, we can identify the slope and y-intercept. The slope m is 4 3 , and the y-intercept b is -3.
Finding the x-intercept To find the x-intercept, we set y = 0 and solve for x : 0 = 4 3 x − 3 . Add 3 to both sides: 3 = 4 3 x . Multiply both sides by 3 4 : x = 3 × 3 4 = 4 . So, the x-intercept is 4.
Conclusion Therefore, we are looking for a graph with a slope of 4 3 , a y-intercept of -3, and an x-intercept of 4.
Examples
Understanding linear equations is crucial in many real-world applications. For instance, consider a taxi service that charges a fixed fee plus a per-mile rate. The equation y = 4 3 x − 3 could represent the total cost ( y ) for a ride of x miles, where -3 is a base charge (perhaps a discount) and 4 3 is the cost per mile. By understanding the slope and intercepts, you can easily calculate the cost of any trip or determine how many miles you can travel for a certain amount of money. This concept extends to budgeting, financial planning, and understanding rates and fees in various services.
