Writing linear equations - standard to slope
Convert the equation from slope-intercept form into the standard form of equation of each line.
1) y =
3
2x
2) y = −x − 3
3) y =
5
2x − 3 4) y =
−1
6x − 5
5) y =
−11
5x + 5
6) y = −2x − 5
-1-
You may use this math worksheet as long as you help someone learn math. -> MATHX.NET <-
7) y =
1
5x − 5 8) y =
−1
3x + 5
9) y = x + 510) y =
−4
3x + 2
11) y = −2x + 212) y =
−2
3x
13) y =
1
6x + 2 14) y =
1
2x − 1
-2-
You may use this math worksheet as long as you help someone learn math. -> MATHX.NET <-
15) y =
3
4x − 3 16) y =
5
4x − 1
17) y =
1
4x − 3
18) y = −4
19) y =
9
2x − 6
20) y = −6
21) x = 4 22) x = −3
-3-
You may use this math worksheet as long as you help someone learn math. -> MATHX.NET <-
23) y = 2x + 4 24) y = −2x + 1
25) y =
−4
3x + 4 26) y =
−1
5x + 1
27) y = −2x − 1 28) y = −3x + 1
29) y = x − 430) y =
7
6x − 2
-4-
You may use this math worksheet as long as you help someone learn math. -> MATHX.NET <-
-5-
Answers to Writing linear equations - standard to slope
1) 3x − 2y = 0 2) x + y = −3 3) 5x − 2y = 6 4) x + 6y = −30
5) 11x + 5y = 25 6) 2x + y = −5 7) x − 5y = 25 8) x + 3y = 15
9) x − y = −5 10) 4x + 3y = 6 11) 2x + y = 2 12) 2x + 3y = 0
13) x − 6y = −12 14) x − 2y = 2 15) 3x − 4y = 12 16) 5x − 4y = 4
17) x − 4y = 12 18) y = −4 19) 9x − 2y = 12 20) y = −6
21) x = 4 22) x = −3 23) 2x − y = −4 24) 2x + y = 1
25) 4x + 3y = 12 26) x + 5y = 5 27) 2x + y = −1 28) 3x + y = 1
29) x − y = 4 30) 7x − 6y = 12
You may use this math worksheet as long as you help someone learn math. -> MATHX.NET <-
-9-
Answers to Writing linear equations - standard to slope
1) 3x + y = 3 2) 2x − y = −4 3) x − 5y = 0 4) x − 2y = 6
5) 6x − 5y = 25 6) 5x + 3y = −6 7) 3x − 4y = 0 8) 3x − 5y = −10
9) x + 4y = 16 10) 10x − y = −5 11) 2x − y = 3 12) x − 2y = 4
13) x + y = −1 14) x − y = −3 15) 5x − 2y = −4 16) x + y = −4
17) 7x + 2y = 10 18) 2x − y = 1 19) 3x − 4y = 12 20) x = −5
21) x + 4y = 4 22) x = 2 23) x = −1 24) 4x − y = 3
25) 7x + y = −2 26) 6x − 5y = −5 27) 7x + 2y = 4 28) 7x − 2y = −6
29) x − 5y = 25 30) 2x − y = 4
You may use this math worksheet as long as you help someone learn math. -> MATHX.NET <-
Lines, Lines, Lines!!! Standard Form of a Linear Equation
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Lines, Lines, Lines!!!
Standard Form of a Linear Equation
~
Lesson Plan
I. Topic: Standard Form
II. Goals and Objectives:
A. The students will convert equations into standard form.
B. The students will write equations in standard form.
C. The students will find the coefficients of the equations.
III. Massachusetts Learning Standards:
1. 8.P.6
Identify the roles of variables within an equation, e.g., y = mx + b, expressing y as a
function of x with parameters m and b.
2. 8.P.7
Set up and solve linear equations and inequalities with one or two variables, using
algebraic methods, models, and/or graph.
3. 10.P.2
Demonstrate an understanding of the relationship between various representations of a
line. Determine a line’s slope and x- and y-intercepts from its graph or from a linear
equation that represents the line. Find a linear equation describing a line from a graph or
geometric description of the line, e.g, by using the “point-slope” or “slope y-intercept”
formulas. Explain the significance of a positive, negative, zero, or undefined slope.
4. 10.P.8
Solve everyday problems that can be modeled using systems of linear equations or
inequalities. Apply algebraic and graphical methods to the solution. Use technology when
appropriate. Include mixture, rate, and work problems.
IV. Materials:
A. Whiteboard with dry-erase markers (Blackboard with chalk could also be used.).
B. Ruler.
C. Pencils.
D. Calculator.
E. Graphing paper.
F. Standard Form Worksheets (Practice Worksheet, Quiz Worksheet).
Lines, Lines, Lines!!! Standard Form of a Linear Equation
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V. Presentation Outline:
A. Introductions. "The Standard Form of a line."
I. Equation
II. Definition
B. Write an equation in Standard Form from the given linear equation.
Example
C. Write an equation in Standard Form given a point and a Slope
Example
D. Write an equation in Standard Form given two points.
Example
VI. Presentation:
A. Presentation Notes
B. Power Point Presentation
VII. Independent Practice: Standard Form Worksheet
A. Class work: # 1 - 29 Odds
B. Homework: #s 2 - 30 Evens
C. Due 2 days from given day. Allow students to complete those questions which they did not
complete in class.
VIII. Topic Assessment: Standard Form Quiz
A. Answer questions from homework.
B. 5 Question Quiz: 10 – 15 minutes
C. 10 Question Quiz: 15 – 20 minutes
Lines, Lines, Lines!!! Standard Form of a Linear Equation
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Lines, Lines, Lines!!!
~
Standard Form of a Linear Equation
Introduction
All of the linear equations we have studied so far can be written in Standard Form.
The Standard form of a linear equation helps us identify its coefficients along with
an easiness of graphing a line by finding the x- and y- intercept of such a line. The
Standard Form is acquired when all of the criteria are met.
Definitions and Formula
The Standard Form of a linear equation is . However, each of the
following criteria must be met:
A, B, and C are integers.
Remember:
Integers are positive and negative numbers including zero which do not have
fractions or decimals. i.e. (...-5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5...).
A, B, and C's greatest common factor is 1.
Remember:
When two numbers have a greater common factor of 1, then they cannot be divided
by any other number than one. i.e. (3 and 5), (10 and 7), (3, 9, and 8), (-4, 11).
'A' is greater than or equal to Zero .
A and B are not both Zero.
Lines, Lines, Lines!!! Standard Form of a Linear Equation
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Key Concepts It is important to learn how to recognize when an equation is in the Standard form
and when it is not.
The following lines are in Standard Form
A. ß Where A = 5, B = -7, and C = 6
B. ß Where A = 1, B = 3, and C = 0
C. ß Where A = 1000, B = 10,000, and C = 1
D. ß Where A = 0, B = 1, and C = 4
The following lines are not in Standard Form:
A. ß 'A' and 'B' must be on the same side.
B. ß 'A' must be greater than or equal to Zero.
C. ß These numbers can be divided by 7.
D. ß is not an integer.
E. 0x + 0y = 4 ß 'A' and 'B' can't both be Zero.
The following lines are in the form of other equations which we will learn or have learned:
A. y = mx + b ß Slope-Intercept Form
B. ß Point-Slope Form
C. Ax + By = C ß Standard Form
D. y = b ß Horizontal Line (Zero Slope)
E. x = b ß Vertical Line (Undefined Slope)
Lines, Lines, Lines!!! Standard Form of a Linear Equation
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Let's see a few ways to use or find the Standard Form of a linear equation.
1. Write an equation in Standard Form from the given linear equation.
A.
i. Original equation.
ii. Add -4x to both sides.
iii. Simplify.
A = 4, B = 1, and C = 8
B.
i. Original equation.
ii. Subtract 8 from both sides.
iii. Simplify.
iv. Add -9y to both sides.
v. Simplify.
A = 3, B = 9, and C = -8
C.
i. Original equation
ii. Multiply all terms by 5.
iii. Simplify.
iv. Subtract 3x from both sides.
Lines, Lines, Lines!!! Standard Form of a Linear Equation
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v. Simplify.
vi. Multiply all terms by -1.
vii. Simplify.
A = 3, B = -5, and C = -25
D.
i. Original equation.
ii. Multiply all terms by 12.
iii. Simplify.
A = 8, B = -3, and C = 72
2. Write an equation in Standard Form given a point and a slope.
Note:
It is always best to used the Point-Slope Form of an equation when the only
information given is one point and the slope of the linear equation.
A. m = and point (3, 4)
i. Point-Slope Form.
ii. Substitute the information given.
iii. Simplify using the Distributive Property.
iv. Multiply all terms by 2.
v. Simplify.
Lines, Lines, Lines!!! Standard Form of a Linear Equation
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vi. Subtract both by sides by x.
vii. Simplify.
viii. Add 8 to both sides.
ix. Simplify.
x. Multiply all terms by -1.
xi. Simplify.
A = 1, B = -2, and C = -5
B. m = 1 and point (-2, 1)
i. Point-Slope Form
ii. Substitute the information given.
iii. Simplify using the Distributive Property.
iv. Subtract both by sides by x.
v. Simplify.
vi. Add 1 to both sides.
vii. Simplify.
viii. Multiply all terms by -1.
ix. Simplify.
A = 1, B = -1, and C = -3
Lines, Lines, Lines!!! Standard Form of a Linear Equation
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C. m = -3 and point (0, -8)
i. Point-Slope Form
ii. Substitute the information given.
iii. Simplify using the Distributive Property.
iv. Simplify.
v. Subtract both by sides by 8.
vi. Simplify.
vii. Add 3x to both sides.
viii. Simplify.
A = 3, B = 1, and C = -8
3. Write an equation in Standard Form given two points.
Note:
(The following examples were used on the previous lesson on Slope-Intercept Form)
A. Write an equation of the line in Standard Form which passes through
(-1, 4) and (1, -2).
i. Find the slope between the two points.
ii. Find the value of the y-intercept (b) by substituting
the value of one of the points into the equation.
iii. Simplify and solve.
iv. Substitute the value of ‘b’ into the equation.
v. Add -3x to both sides.
vi. Simplify.
A = 3, B = 1 and C = 1
Lines, Lines, Lines!!! Standard Form of a Linear Equation
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B. Write an equation of the line in Standard Form which passes through
(5, -3) and (-4, 3).
i. Find the slope between the two points.
ii. Find the value of the y-intercept (b) by substituting
the value of one of the points into the equation.
iii. Simplify and solve.
iv. Substitute the value of ‘b’ into the equation.
v. Add to both sides.
vi. Simplify.
vii. Multiply all terms by 3.
viii. Simplify.
A = 3, B = 3 and C = 1
C. Write an equation of the line in Standard Form which passes through
(-1, 2) and (3, 4).
i. Find the slope between the two points.
ii. Find the value of the y-intercept (b) by substituting
the value of one of the points into the equation.
iii. Simplify and solve.
Lines, Lines, Lines!!! Standard Form of a Linear Equation
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iv. Substitute the value of ‘b’ into the equation.
v. Subtract from both sides.
vi. Simplify.
vii. Multiply all terms by -2.
viii. Simplify.
A = 1, B = -2 and C = -5
Lines, Lines, Lines!!! Standard Form of a Linear Equation
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Standard Form of a Line ~
Student Practice Worksheet
Name____________________________________________Date______________Grade___________
Write an equation in Standard Form given a line in slope-intercept form.
1. y = 2. y =
3. y = 4. y =
5. y = 6. y =
7. y = 8. y =
9. y = 10. y =
Write an equation in Standard Form given two points.
11. (-3, 1) and (5, 4) 12. (5, -6) and (2, 3)
Lines, Lines, Lines!!! Standard Form of a Linear Equation
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13. (2, -2) and (-6, 1) 14. (3, 4) and (-7, 4)
15. (0, -2) and (7, 0) 16. (-5, -1) and (4, -7)
\
17. (1, 0) and (4, 1) 18. (-2, 0) and (0, -2)
19. (-1, -2) and (1, 4) 20. (-2, 5) and (-1, 2)
Write an equation in Standard Form given a point and a Slope.
21. (2, 3), m = 3 22. (-4, 7), m = 3
23. (0, 3), m = -4 24. (7, 2), m = -5
25. (3, 2), m = 26. (2, -3), m =
27. (1, 2), m = 7 28. (3, -1), m = -1
29. (-2, 5), m = -4 30. (3, 5), m =
Lines, Lines, Lines!!! Standard Form of a Linear Equation
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Standard Form of a Line
~
Student Practice Worksheet
Answer Key
Name______________________________________________ Date______________ Grade_________
Write an equation in Standard Form given a line in slope-intercept form.
1. y = 2. y =
7x – 10y = -8 4x + 5y = -5
3. y = 4. y =
4x + 10y = 3 7x + 7y = 4
5. y = 6. y =
3x + 7y = -4 x – 5y = - 5
7. y = 8. y =
6x + 6y = 1 2x + 10y = -7
9. y = 10. y =
2x – 5y = 2 4x – 7y = -5
Write an equation in Standard Form given two points.
11. (-3, 1) and (5, 4) 12. (5, -6) and (2, 3)
3x – 8y = -17 3x + y = 9
Lines, Lines, Lines!!! Standard Form of a Linear Equation
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(Standard Form Worksheet answer key)
13. (2, -2) and (-6, 1) 14. (3, 4) and (-7, 4)
3x + 8y = -10 y = 4
15. (0, -2) and (7, 0) 16. (-5, -1) and (4, -7)
2x – 7y = 14 2x + 3y = - 7
17. (1, 0) and (4, 1) 18. (-2, 0) and (0, -2)
x – 3y = 1 x + y = -2
19. (-1, -2) and (1, 4) 20. (-2, 5) and (-1, 2)
3x – y = - 1 3x + y = -1
Write an equation in Standard Form given a point and a Slope.
21. (2, 3), m = 3 22. (-4, 7), m = 3
3x – y = 3 3x – y = -19
23. (0, 3), m = -4 24. (7, 2), m = -5
4x + y = 3 5x + y = 37
25. (3, 2), m = 26. (2, -3), m =
2x – 3y = 0 3x + 2y = 0
27. (1, 2), m = 7 28. (3, -1), m = -1
7x – y = 5 x + y = 2
29. (-2, 5), m = -4 30. (3, 5), m =
4x + y = -3 5x – 3y = 0
Lines, Lines, Lines!!! Standard Form of a Linear Equation
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Standard Form of a Line
~
Student Practice Worksheet
Rubric
Criteria
4 3 2 1 0
Mechanics
No math errors
No major math
errors or serious
flaws in reasoning.
May be some
serious math
error or flaws in
reasoning.
Major math errors
or serious flaws
in reasoning.
Blank
answers
Quiz Grading Rubric:
Problem Total points of
Correct
Answer
Problem Total points of
Correct
Answer
1. 16.
2. 17.
3. 18.
4. 19.
5. 20.
6. 21.
7. 22.
8. 23.
9. 24.
10. 25.
11. 26.
12. 27.
13. 28.
14. 29.
15. 30.
Total: _______________
Lines, Lines, Lines!!! Standard Form of a Linear Equation
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Standard Form of a Line
~
Quiz
Name____________________________________________Date_______________Grade___________
Write an equation in Standard Form given a line in slope-intercept form.
1. y = -2x – 18 2.
3. 10 – 2x = 8y 4. y =
5. y =
Write an equation in Standard Form given a point and a slope.
6. (2, -4), slope = -1 7. (3, 1), slope = ½
8. (-1, 2), slope = 2 9. (-4, 0), slope = ¾
10. (-3, -3), slope =
Lines, Lines, Lines!!! Standard Form of a Linear Equation
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Write an equation in Standard Form given two points.
11. (4, -2) and (8, 3) 12. (2, 10) and (8, 7)
13. (7, 3) and (8, 5) 14. (5, 7) and (8, 5)
15. (10, 3) and (7, 9)
Lines, Lines, Lines!!! Standard Form of a Linear Equation
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Standard Form of a Line
~
Quiz
Answer Key
Name____________________________________________Date_______________Grade___________
Write an equation in Standard Form given a line in slope-intercept form.
1. y = -2x – 18 2.
2x + y = -18 2x + 15y = 50
3. 10 – 2x = 8y 4. y =
x + 4y = 5 x – 7y = 7
5. y =
5x + y = -4
Write an equation in Standard Form given a point and a slope.
6. (2, -4), slope = -1 7. (3, 1), slope = ½
x + y = -2 x – 2y = 1
8. (-1, 2), slope = 2 9. (-4, 0), slope = ¾
2x – y = 4 3x – 4y = -12
10. (-3, -3), slope =
7x – 3y = - 12
Lines, Lines, Lines!!! Standard Form of a Linear Equation
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(Standard Form Quiz Answer Key continue)
Write an equation in Standard Form given two points.
11. (4, -2) and (8, 3) 12. (2, 10) and (8, 7)
5x – 4y = 28 x + 2y = 22
13. (7, 3) and (8, 5) 14. (5, 7) and (8, 5)
2x – y = 11 2x + 3y = 31
15. (10, 3) and (7, 9)
2x + y = 23
Lines, Lines, Lines!!! Standard Form of a Linear Equation
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Standard Form of a Line
~
Quiz
Rubric
Criteria
4 3 2 1 0
Mechanics
No math errors
No major math
errors or serious
flaws in reasoning.
May be some
serious math
error or flaws in
reasoning.
Major math errors
or serious flaws
in reasoning.
Blank
answers
Quiz Grading Rubric:
Problem Total points of
Correct
Answer
Problem Total points of
Correct
Answer
1. 9.
2. 10.
3. 11.
4. 12.
5. 13.
6 14.
7 15.
8
Total:_________
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I. Model Problems. II. Practice
III. Challenge Problems
VI. Answer Key
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I. Model Problems
The “standard” form of a line is given by the formula Ax + By = C,
where A and B are both not equal to zero.
Standard form is useful for finding x- and y-intercepts.
Example 1 What is the x-intercept of the line with equation 3x + 4y =
12?
3x + 4y = 12 Write the equation of the line.
3x + 4(0) = 12 Substitute y = 0 to find the x-
intercept.
3x = 12 Simplify.
x = 4 Divide.
The x-intercept is 4, or the point (4, 0).
The answer is (4, 0).
Example 2 What is the y-intercept of the line with equation 3x + 4y =
12?
3x + 4y = 12 Write the equation of the line.
3(0) + 4y = 12 Substitute x = 0 to find the y-
intercept.
4y = 12 Simplify.
y = 3 Divide.
The y-intercept is 3, or the point (0, 3).
The answer is (0, 3).
The x-intercept of a line in form Ax + By = C is equal to C/A.
The y-intercept of a line in form Ax + By = C is equal to B/A.
Equations written in standard form are typically written using integer
coefficients for x and y. This means the values of A and B are usually
integers.
Converting to standard form from slope-intercept form is a useful skill.
Example 3 Write y =
3
2x + 10 in standard form using integer
coefficients.
y =
3
2x + 10
Write the equation of the line.
-
3
2x + y = 10 Subtract
3
2x to get the x and y on
the same side of the equation.
2•( -
3
2x + y) = 2•10
Multiply each side by 2.
–3x + 2y = 20 Simplify.
The answer is –3x + 2y = 20.
Sometimes you will need to graph a line in standard form.
Example 4 Graph the line with equation 2x + 8y = 16.
Calculate the x- and y- intercepts as shown in Example 1 and Example
2:
2x + 8y = 16 Write the equation of the line.
2x + 8(0) = 16 Substitute y = 0 to find the x-
intercept.
2x = 16 Simplify.
x = 8 Divide.
The x-intercept is (8, 0).
2x + 8y = 16 Write the equation of the line.
2(0) + 8y = 16 Substitute x = 0 to find the y-
intercept.
8y = 16 Simplify.
y = 2 Divide.
The y-intercept is (0, 2).
Plot the points (8, 0) and (0, 2). Draw the line between the two points.
The graph is shown:
II.
Practice
Find the x- and y- intercepts of the given lines in standard form.
1. 2x + 5y = 10 2. 3x + 8y = 24
3. -2x +7y = 14 4. -9x – 2y = 18
5. 4x – 5y = 8 6. 25x + 100y = 200
7. 8x + 11y = 92 8. 12x – 36y = 72
9. -70x + 90y = 630
10. -10x + 5y = 45
Write each equation in standard form using integer coefficients for A, B
and C.
11. y =
3
2x + 10 12. y =
-
1
4x + 2
13. y = -
7
3x + 5
14. y = 5x + 8
15. y =
5
6x – 7 16. y =
1
3x –
2
3
17. y = -
8
15x +
1
20 18. y =
2
5x +
5
6
19. y = -
3
8x +
2
15
20. y = -
6
7x +
1
9
21. y = -
14
17x +
15
17
22. y =
1
12x +
1
3
Graph each line using intercepts.
23. 3x + 5y = 15 24. 7x – 3y = 21
25. 8x + 7y = 56 26. -5x – 4y = 40
27. -6x + 9y = 54 28. 9x – 10y = 90
29. 5x – 7y = 42 30. -5x – 2y = 18
31. 9x + 11y = 88
32. 10x – 6y = 35
33. 12x + 5y = 144 34. 15x – 30y = 60
III. Challenge Problems
35. Explain why it’s sometimes helpful to use the standard form instead
of the slope-intercept form of a line.
_________________________________________________________
_________________________________________________________
36. Write y – j = m(x – k) in standard form.
_________________________________________________________
37. Correct the Error
Question: Write y = -
1
5x +
3
4 in standard form using integer
coefficients for A, B and C.
Solution:
y =
-1
5x +
3
4
y +
1
5x =
3
4
5y + x =
3
4
5y + 4x = 3, or 4x + 5y =3 The equation of the line is 4x + 5y = 3.
What is the error? Explain how to solve the problem.
_________________________________________________________
_________________________________________________________
IV. Answer Key
1. x-intercept = (5, 0); y-intercept = (0, 2)
2. x-intercept = (8, 0); y-intercept = (0, 3)
3. x-intercept = (-7, 0); y-intercept = (0, 2)
4. x-intercept = (-2, 0); y-intercept = (0, -9)
5. x-intercept = (2, 0); y-intercept = (0, -8/5)
6. x-intercept = (8, 0); y-intercept = (0, 2)
7. x-intercept = (92/8, 0); y-intercept = (0, 92/11)
8. x-intercept = (6, 0); y-intercept = (0, -2)
9. x-intercept = (-9, 0); y-intercept = (0, 7)
10. x-intercept = (-45/10, 0); y-intercept = (0, 9)
11. -3x + 2y = 20
12. x + 4y = 8
13. 7x + 3y = 15
14. -5x + y = 8
15. 5x – 6y = 42
16. x – y = 2
17. 160x + 300y = 15
18. 12x – 30y = -25
19. 45x + 90y = 16
20. 54x + 63y = 1
21. 14x + y = 15
22. x – 12y = -4
23.
24.
25.
26.
27.
28.
29.
30.
31.
32.
33.
34.
35. The standard form makes it easier to calculate the x- and y-
intercepts.
36. mx – y = mk – j
37. The student did not multiply the 4 each term in the last line; the
correct answer is 4x + 20y = 3.