# Linear Equations Worksheet-Solutions

1. Given is the line with equation y = 3x − 2.

(a) Find five points on the line and arrange them in a table.

**Answer**

y = 3x − 2 | |

x | y |

-2 | -8 |

0 | -2 |

1 | 1 |

3 | 7 |

10 | 28 |

(b) Graph the line .

**Answer **See the graph on the last page.

(c) Find the x- intercept and the y -intercept.

**Answer **To find the x-intercept let y = 0 in y = 3x
− 2 and solve for x.

Thus, the x- intercept is the point ( 2/3 , 0).

To find the y-intercept let x = 0 in y = 3x − 2 and solve for y .

Thus, the y-intercept is the point (0,−2).

2. Find the slope-intercept form of the equation of the
line through the points (2, 7)

and (5, 2) and graph it .

**Answer **First calculate the slope.
So far we have

To solve for b we substitute the coordinates of a point on
the line, for example (2, 7).

Then at the point (2, 7) we have

and so the answer is

We should check our work by verifying that the other point
also lies on the line.

In other words, substituting the point (5, 2) we should obtain an identity.

3. Consider the line passing through the point (2, 3) with slope m = −1.

(a) Write down the point-slope equation of the line.

**Answer**

y − 3 = −(x − 2)

(b) Write the equation in the slope-intercept form.

**Answer**

y = −x + 5

(c) Find all intercepts.

**Answer **The x-intercept is the point (5, 0) and the
y-intercept is the point (0, 5).

4. Consider the line y = 2x + 3.

(a) Find the equation in slope-intercept form of a parallel line through (2, 5).

**Answer** The given line has slope m = 2 so we are
looking for a line of the

form y = 2x + b and containing the point (2, 5). Substituting x = 2 it follows

that b = 1 in order for y = 5. Thus, we obtain

y = 2x + 1

(b) Find the equation of a perpendicular line through (2, 7).

**Answer **The line has slope m = 2 so
and a perpendicular line

will have the form
Substituting the point (2, 7) and solving for b

we obtain

5. Consider the line L given by 2x + 3y = 6.

(a) Find the slope and intercepts of the line.

**Answer** In the slope-intercept form we have
so the slope is

m = −2/3 . The x-intercept is the point (3, 0) and the y-intercept is the point
(0, 2).

(b) Find a point on the line and a point not on the line.

**Answer **(0, 0) does not lie on the line, but (3, 0)
does.

(c) Write the equation of the line in point-slope form.

**Answer **The slope we already know to be m = −2/3 and
we can choose the

point (3, 0), so

(d) Find the equation of a line perpendicular to L, but
passing through the same

x-intercept as the line L.

**Answer** We have m = −2/3 so
In slope-intercept form we have

and we need to have this line pass through the point (3,
0). Substituting we

find that b = −9/2 and so the answer is

6. Solve:

**Answer** Proceed by elimination: rewrite the system
of equations and add them.

We have

and whence −4y = 9
y = −9/4 . Then we substitute y = −9/4 into the first equation

and solve for x and obtain x = −5/8 . To make sure that
is the solution we

check that it also solves the second equation:

7. Derive the point-slope form of the equation for a line by following these steps .

(a) Let L be the line passing through the fixed point
(x_{1}, y_{1}) and an arbitrary point

(x, y).

(b) Find the general formula for the slope of L.

**Answer** The slope of L is given by
Multiplying thru by (x − x_{1}) we

obtain the point-slope form.

8. *Write down a system of 3 linear equations that has

(a) exactly one solution

**Answer **All the above problems have exactly one solution. Take for example

problem 6 and introduce a third line which passes through the solution

We use the slope-intercept form with an arbitrary slope, say m = 2.

(b) no solution

**Answer **The only three lines in the plane that do not intersect are parallel

lines. We can take for example the line 2x − 5y = 10 and pick 3 different
y -intercepts.

(c) infinitely many solutions

**Answer** Infinitely many solutions occur when the three lines are in fact the

same line. That is, we have three parallel lines with the same y-intercept.

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