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.

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