Linear Functions - Math

First, we need to find the slope of the line.

In order to move from the lower left point to the upper right point, it is necessary to move up five units and right three units. This is a rise of 5 and a run of 3. makes the slope of the line shown .

We can use this to find the -intercept using the slope formula as follows:

The lower left point has coordinates . Therefore, we can set up and solve for in this slope formula, setting :

We can solve the system of equations using the substitution method.

Solve for in the second equation.

Substitute this value of into the first equation.

Now we can solve for .

Solve for using the first equation with this new value of .

The solution is the ordered pair .

Let's start by looking at the given equation:

The inequality is written in slope-intercept form; therefore, the slope is equal to and the y-intercept is equal to .

All of the graphs depict a line with slope of and y-intercept . Next, we need to decide if we should shade above or below the line. To do this, we can determine if the statement is true using the origin . If the origin satisfies the inequality, we will know to shade below the line. Substitute the values into the given equation and solve.

Because this statement is true, the origin must be included in the shaded region, so we shade below the line.

Finally, a statement that is "less than" or "greater than" requires a dashed line in the graph. On the other hand, those that are "greater than or equal to" or "less than or equal to" require a solid line. We will select the graph with shading below a dashed line.

A horizontal line has equation for some value of ; since the line goes through a point with -coordinate 3, the line is . Also, since the line is solid and the region above this line is shaded in, the corresponding inequality is .

A vertical line has equation for some value of ; since the line goes through a point with -coordinate 4, the line is . Also, since the line is solid and the region right of this line is shaded in, the corresponding inequality is .

Since only the region belonging to both sets is shaded - that is, their intersection is shaded - the statements are connected with "and". The correct choice is .

First, we determine the equation of the boundary line. This line includes points and , so the slope can be calculated as follows:

Since we also know the -intercept is , we can substitute in the slope-intercept form to obtain the equation of the boundary line:

The boundary is included, as is indicated by the line being solid, so the equality symbol is replaced by either or . To find out which one, we can test a point in the solution set - for ease, we will choose :

0 is less than 3 so the correct symbol is .

The inequality is .

To find an equation of a line, we will always need to know the slope of that line -- and to find the slope, we need at least two points. It looks like we have (0, -3) and (12,0), which we'll call point 1 and point 2, respectively.

Now we need to plug in a point on the line into an equation for a line. We can use either slope-intercept form or point-slope form, but since the answer choices are in point-slope form, let's use that.

Unfortunately, that's not one of the answer choices. That's because we didn't pick the same point to substitute into our equation as the answer choices did. But we can see if any of the answer choices are equivalent to what we found. Our equation is equal to:

which is the slope-intercept form of the line. We have to put all the other answer choices into slope-intercept to see if they match. The only one that works is this one:

One way to answer this is to first find the equation of the line.

The slope of a line. given two points can be calculated using the slope formula

Set :

The line has slope 3 and -intercept , so we can substitute in the slope-intercept form:

Now substitute 4 for and for and solve for :

First, we need to find the slope of the above line.

The slope of a line. given two points can be calculated using the slope formula

Set :

The slope of a line perpendicular to it has as its slope the opposite of the reciprocal of 2, which would be . Since we want this line to have the same -intercept as the first line, which is the point , we can substitute and in the slope-intercept form:

We calculate the slopes of the lines using the slope formula.

The slope of line is

The slope of line is

The lines have the same slope, making them either parallel or identical.

Since the slope of each line is 0, both lines are horizontal, and the equation of each takes the form , where is the -coordinate of each point on the line. Therefore, line and line have equations and .This makes them parallel lines.

In form, where y = maximum heart rate and x = age, we can express the relationship as:

We are looking for a graph with a slope of -1 and a y-intercept of 220.

The slope is -1 because as you grow one year older, your maximum heart rate decreases by 1.

Lines are perpendicular when their slopes are the negative recicprocals of each other such as . To find the slope of our equation we must change it to slope y-intercept form.

Subtract the x variable from both sides:

Divide by 4 to isolate y:

The negative reciprocal of the above slope: . The only equation with this slope is .

A line tangent to a circle at a given point is perpendicular to the radius from the center to that point. That radius, which has endpoints , has slope

.

The line, being perpendicular to this radius, will have slope equal to the opposite of the reciprocal of that of the radius. This slope will be . Since it includes point , we can use the point-slope form of the line to find its equation:

Use substution to solve this problem:

becomes and then is substituted into the second equation. Then solve for :

, so and to give the solution .

To solve for the -intercept, set to zero and solve for :

To solve for the -intercept, set to zero and solve for :

Slope-intercept form is .

A vertical line has infinitely many values of but only one value of . Thus, vertical lines are of the form , where is a real number. The only equation of this form is .

A horizontal line has infinitely many values for , but only one possible value for . Thus, it is always of the form , where is a constant. Horizontal lines have a slope of . The only equation of this form is .

A vertical line is one in which the values can vary. Namely, there is only one possible value for , and can be any number. Thus, by this description, the only vertical line listed is .

A line with a slope of zero will be horizontal. A horizontal line has only one possible value for , and can be any value.

Thus, the only given equation which fits this description is .

The equation of a line is , with m being the slope of the line, and b being the y-intercept. The y intercept of the line is at , so .

The x-intercept is at , the equation becomes , simplification yields