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Linear Equations in Two Variables
In this chapter, we’ll use the geometry of lines to help us solve equations.
Linear equations in two variables
If a, b, and r are real numbers (and if a and b are not both equal to 0) then ax + by = r is called a linear equation in two variables. (The “two variables” are the x and the y.) The numbers a and b are called the coe cients of the equation ax + by = r. The number r is called the constant of the equation ax + by = r.
Examples. 10 x 3 y = 5 and 2 x 4 y = 7 are linear equations in two variables.
Solutions of equations
A solution of a linear equation in two variables ax+by = r is a specific point in R 2 such that when when the x-coordinate of the point is multiplied by a, and the y-coordinate of the point is multiplied by b, and those two numbers are added together, the answer equals r. (There are always infinitely many solutions to a linear equation in two variables.)
Example. Let’s look at the equation 2x 3 y = 7. Notice that x = 5 and y = 1 is a point in R 2 that is a solution of this equation because we can let x = 5 and y = 1 in the equation 2x 3 y = 7 and then we’d have 2(5) 3(1) = 10 3 = 7. The point x = 8 and y = 3 is also a solution of the equation 2x 3 y = 7 since 2(8) 3(3) = 16 9 = 7. The point x = 4 and y = 6 is not a solution of the equation 2x 3 y = 7 because 2(4) 3(6) = 8 18 = 10, and 10 6 = 7.
To get a geometric interpretation for what the set of solutions of 2x 3 y = 7 looks like, we can add 3y, subtract 7, and divide by 3 to rewrite 2x 3 y = 7 as 23 x 73 = y. This is the equation of a line that has slope 23 and a y-intercept of 73. In particular, the set of solutions to 2x 3 y = 7 is a straight line. (This is why it’s called a linear equation.)
Linear equations and lines
If b = 0, then the linear equation ax + by = r is the same as ax = r. Dividing by a gives x = ra , so the solutions of this equation consist of the points on the vertical line whose x-coordinates equal (^) ar.
If b 6 = 0, then the same ideas from the 2x 3 y = 7 example that we looked at previously shows that ax + by = r is the same equation as, just written in a di↵erent form from, ab x + rb = y. This is the equation of a straight line whose slope is ab and whose y-intercept is rb.
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Linear equations and lines.
If b = 0, then the linear equation ax + by = r is the same as ax = r. Dividing by a gives x = r a , so the solutions to this equation consist of the points on the vertical line whose x-coordinates equal r a.
If b ⇥= 0, then the same ideas from the 2x 3 y = 7 example that we looked at previously shows that ax + by = r is the same equation as, just written in a di erent form from, a b x + r b = y. This is the equation of a straight line whose slope is a b and whose y-intercept is r b.
Linear equations and lines.
If b = 0, then the linear equation ax + by = r is the same as ax = r. Dividing by a gives x = (^) ar , so the solutions to this equation consist of the points on the vertical line whose x-coordinates equal r a.
If b 6 = 0, then the same ideas from the 2x 3 y = 7 example that we looked at previously shows that ax + by = r is the same equation as, just written in a di erent form from, a b x + r b = y. This is the equation of a straight line whose slope is a b and whose y-intercept is r b.
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Linear equations and lines.
If b = 0, then the linear equation ax + by = r is the same as ax = r. Dividing by a gives x = r a , so the solutions to this equation consist of the points on the vertical line whose x-coordinates equal r a.
If b ⇥= 0, then the same ideas from the 2x 3 y = 7 example that we looked at previously shows that ax + by = r is the same equation as, just written in a di erent form from, a b x + r b = y. This is the equation of a straight line whose slope is a b and whose y-intercept is r b.
186
Linear equations and lines.
If b = 0, then the linear equation ax + by = r is the same as ax = r. Dividing by a gives x = r a , so the solutions to this equation consist of the points on the vertical line whose x-coordinates equal r a.
If b 6 = 0, then the same ideas from the 2x 3 y = 7 example that we looked at previously shows that ax + by = r is the same equation as, just written in a di erent form from, a b x + r b = y. This is the equation of a straight line whose slope is a b and whose y-intercept is r b.
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The first equation in this system, 8x + 7y = 38, corresponds to a line that has slope 87. The second equation in this system, 3x 5 y = 3, is represented by a line that has slope (^) ^35 = 35. Since the two slopes are not equal, the lines have to intersect in exactly one point. That one point will be the unique solution. As we’ve seen before, x = 3 and y = 2 is a solution of this system. It is the unique solution.
Example. The system
5 x + 2y = 4 2 x + y = 11
has a unique solution. It’s x = 2 and y = 7. It’s straightforward to check that x = 2 and y = 7 is a solution of the system. That it’s the only solution follows from the fact that the slope of the line 5x + 2y = 4 is di↵erent from slope of the line 2 x + y = 11. Those two slopes are 52 and 2 respectively.
No solutions
If you reach into a hat and pull out two di↵erent linear equations in two variables, it’s highly unlikely that the two corresponding lines would have exactly the same slope. But if they did have the same slope, then there 247
The first equation in this system, 8x + 7y = 38, corresponds to a line that has slope ^87. The second equation in this system, 3x 5 y = 3, is represented by a line that has slope (^) ^35 = 35. Since the two slopes are not equal, the lines have to intersect in exactly one point. That one point will be the unique solution. As we’ve seen before that x = 3 and y = 2 is a solution to this system, it must be the unique solution.
Example. The system 5 x + 2y = 4 2 x + y = 11 has a unique solution. It’s x = 2 and y = 7. It’s straightforward to check that x = 2 and y = 7 is a solution to the system. That it’s the only solution follows from the fact that the slope of the line 5x + 2y = 4 is di erent from slope of the line 2 x + y = 11. Those two slopes are ^52 and 112 respectively.
No solutions.
If you reach into a hat and pull out two di erent linear equations in two variables, it’s highly unlikely that the two corresponding lines would have exactly the same slope. But if they did have the same slope, then there 188
The first equation in this system, 8x + 7y = 38, corresponds to a line that has slope ^87. The second equation in this system, 3x 5 y = 3, is represented by a line that has slope (^) ^35 = 35. Since the two slopes are not equal, the lines have to intersect in exactly one point. That one point will be the unique solution. As we’ve seen before that x = 3 and y = 2 is a solution to this system, it must be the unique solution.
Example. The system 5 x + 2y = 4 2 x + y = 11 has a unique solution. It’s x = 2 and y = 7. It’s straightforward to check that x = 2 and y = 7 is a solution to the system. That it’s the only solution follows from the fact that the slope of the line 5x + 2y = 4 is di erent from slope of the line 2 x + y = 11. Those two slopes are ^52 and 112 respectively.
No solutions.
If you reach into a hat and pull out two di erent linear equations in two variables, it’s highly unlikely that the two corresponding lines would have exactly the same slope. But if they did have the same slope, then there
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would not be a solution of the system of two linear equations since no point in R 2 would lie on both of the parallel lines.
Example. The system
x 2 y = 4 3 x + 6y = 0
does not have a solution. That’s because each of the two lines has the same slope, 12 , so the lines don’t intersect.
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would not be a solution to the system of two linear equations since no point in R^2 would lie on both of the parallel lines.
Example. The system x 2 y = 4 3 x + 6y = 0 does not have a solution. That’s because each of the two lines has the same slope, 12 , so the lines don’t intersect.
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would not be a solution to the system of two linear equations since no point in R^2 would lie on both of the parallel lines.
Example. The system x 2 y = 4 3 x + 6y = 0 does not have a solution. That’s because each of the two lines has the same slope, 12 , so the lines don’t intersect.
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14.) Is there a unique solution to the system
6 x + 2y = 4 15 x + 5y = 7
For #15-17, find the roots of the given quadratic polynomials.
15.) 9x 2 12 x + 4
16.) 2x 2 3 x + 1
17.) 4 x 2 + 2x 3
