Chapter Six — Linear Equations in Two Variables
The Humongous Book of Algebra Problems
106
Point-Slope Form of a Linear Equation
Point + slope = equation
6.1 Identify the values of the constants in the point-slope form of a line.
The point-slope form of a linear equation states that a line with slope m that
passes through point (x
1
,y
1
) has equation y – y
1
= m(x – x
1
). Any point (x,y) that
satisfies the equation lies on the graph of the line.
6.2 Describe the difference between the subscripted variables in the point-slope
form of a line (x
1
and y
1
) and the variables without subscripts (x and y).
The point-slope form of a line is used to create the equation of a line if the
slope of that line and one of the points on that line are known. The coordinates
(x
1
,y
1
) represent the x- and y-values of the point through which the line is known
to pass. The point-slope formula also contains the variables x and y, which
correspond to any point (x,y) through which the line passes.
Note: Problems 6.3–6.4 refer to line j, which has slope –4 and passes through the point
(–1,7) on the coordinate plane.
6.3 Use the point-slope formula to write an equation representing line j.
Line j has slope m = –4 and passes through the point (–1,7), so x
1
= –1 and y
1
= 7.
Substitute the values of m, x
1
, and y
1
into the point-slope formula.
Note: Problems 6.3–6.4 refer to line j, which has slope –4 and passes through the point
(–1,7) on the coordinate plane.
6.4 If line j also passes through the point (a,3), what is the value of a?
If line j passes through the point (a,3), then substituting x = a and y = 3 into the
equation y – 7 = –4(x + 1) must result in a true statement.
x
1
and y
1
are only needed
at the beginning of
the problem, when you
create the equation.
The equation you come
up with will not contain
x
1
and y
1
, but it will
contain x and y.
Substitute into the
equation of line j from
Problem 6.3.
Chapter Six — Linear Equations in Two Variables
The Humongous Book of Algebra Problems
107
Solve the equation for a.
Because a = 0, line j passes through the point (a,3) = (0,3).
Note: Problems 6.5–6.6 refer to line k, which has slope and passes through the point
(–6,–5) on the coordinate plane.
6.5 Use the point-slope formula to create the equation of line k. Expand the
resulting equation and solve it for y.
Substitute , x
1
= –6, and y
1
= –5 into the point-slope formula.
Expand the right side of the equation and solve for y.
Note: Problems 6.5–6.6 refer to line k, which has slope and passes through the point
(–6,–5) on the coordinate plane.
6.6 If line k also passes through the point , what is the value of c?
Substitute and y = c into the equation generated by Problem 6.5.
Treat the
negatives outside
these parentheses
as –1s: (–1)(–5) = 5
and –1(–6) = 6.
A negative times a
negative equals a
positive.
Use the least common denominator
to combine these numbers:
Chapter Six — Linear Equations in Two Variables
The Humongous Book of Algebra Problems
108
6.7 Use the point-slope formula to create the equation of line l, which has slope
and x-intercept 1.
If line l has x-intercept 1, it passes through the point (1,0). Substitute ,
x
1
= 1, and y
1
= 0 into the point-slope formula.
Line l has equation . Applying the distributive property produces
another valid representation of line l: .
6.8 Use the point-slope formula to identify the equation of the line that passes
through points (3,–1) and (–7,–9). Expand the resulting equation, simplify it,
and solve for y.
Apply the technique described in Problems 5.28–5.30 to calculate the slope of
the line; substitute x
1
= 3, y
1
= –1, x
2
= –7, and y
2
= –9 into the slope formula.
Substitute , x
1
= 3, and y
1
= –1 into the point-slope formula.
Expand and simplify the right side of the equation.
Solve for y and use a common denominator to combine the constant terms.
For more
info on x- and y-
intercepts, check out
Problems 5.18–5.25.
Apply the
distributive
property and simplify.
You could
substitute the
coordinates of
the other point into
the formula instead:
x
1
= –7 and y
1
= –9.
Either way, you’ll
get the same
nal answer.
Chapter Six — Linear Equations in Two Variables
The Humongous Book of Algebra Problems
109
6.9 Use the point-slope formula to identify the equation of the line with x-
intercept 1 and y-intercept –9. Expand the resulting equation, simplify it, and
solve for y.
If the line has x-intercept 1, then it passes through the point (1,0). Similarly,
because the line has y-intercept –9, it passes through the point (0,–9). Calculate
the slope of the line.
Substitute m = 9, x
1
= 1, and y
1
= 0 into the point-slope formula.
6.10 Use the point-slope formula to identify the equation of the line that passes
through the points (a,b) and (c,d). Assume that a, b, c, and d are real numbers
and that a ≠ c.
Apply the technique described in Problems 6.8–6.9. Begin by substituting x
1
= a,
y
1
= b, x
2
= c, and y
2
= d into the slope formula.
Substitute m, x
1
, and y
1
into the point-slope formula.
Substituting (x
1
,y
1
) = (c,d) into the point-slope formula, rather than using the
coordinate (a,b), produces the equally valid equation .
If you set
x
1
= c, y
1
= d,
x
2
= a, and y
2
= b,
you get a different-
looking (but equal)
slope. You can use
that slope to come up
with two different (and
equivalent) equations
that are also acceptable
answers:
and
.
x
1
= 1,
y
1
= 0, x
2
= 0,
and y
2
= –9.
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