Which linear function represents the line given by the point-slope equation y + 7 = –(x + 6)?
- a) f(x) = x-11
- b) f(x) = -x-1
- c) f(x) = -x +3
- d) f(x) = -x -13
The correct answer is d) f(x) = -x -13
There are a few ways to solve this equation, but one common method is to use inverse operations to isolate the variable on one side. Here’s one possible solution:
Start by distributing the negative sign on the right side: y + 7 = -x – 6
Next, add x to both sides: y + x + 7 = -6
Then, add 6 to both sides: y + x + 13 = 0
Finally, combine like terms: y + x = -13
So the solution is y + x = -13. This means that any point that satisfies this equation is on the line y + x = -13. For example, (x, y) = (-3, -10) is a point on this line because -3 + -10 = -13.
Another way to interpret this solution is to think of it as a slope-intercept form of a linear equation. In slope-intercept form, the equation is represented as y = mx + b, where m is the slope of the line and b is the y-intercept.
In this case, the slope of the line is 1, because the coefficient of x is 1. The y-intercept is -13, because the constant term is -13.
We can also represent this equation in point-slope form, which is represented as y – y1 = m(x – x1). In this form, the slope is represented by m, (x1,y1) is a point on the line and m represents the slope of the line.
Another way to visualize this equation is by graphing it on a coordinate plane. We can plot the point (-13,0) on the y-axis, which is the y-intercept of the line. Then, we can use the slope of 1 to find a second point by going up 1 unit on the y-axis and right 1 unit on the x-axis. The two points we have now are (-13,0) and (-12,1). We can use these two points to graph the line on the coordinate plane.
In summary, the equation y + x = -13 can be interpreted as a slope-intercept form, point-slope form and a graph. Each representation gives us a different perspective on the line represented by this equation.
The linear function y = mx + b is a function whose slope is m and its y-intercept is b. A function’s slope determines its rate of change, while its y-intercept indicates where it crosses the y-axis.
A linear equation can be written in point-slope form by using a point on the line and its slope. Point-slope equations are expressed as y – y1 = m(x – x1), where (x1, y1) represents the point on the line and m represents the slope. By using this form, we can determine the equation of a line given a point and a slope, or we can determine the slope of a line given two points on it.