Solve the system of equations.
9x + y = 2
–4x – y = –17
I don't understand how you solve this.
Add the two equations to cancel out y.
5x = -15
You should be able to work it from there.
9 x + y = 2 Subtract 9 x to both sides
9 x + y - 9 x = 2 - 9 x
y = 2 - 9 x
–4 x – y = – 17
–4 x – ( 2 - 9 x ) = – 17
- 4 x - 2 - ( - 9 x ) = - 17
- 4 x - 2 + 9 x = - 17
- 4 x + 9 x - 2 = - 17
5 x - 2 = - 17 Add 2 to both sides
5 x - 2 + 2 = - 17 + 2
5 x = - 15 Divide both sides by 5
5 x / 5 = - 15 / 5
x = - 3
y = 2 - 9 x = 2 - 9 * ( - 3 ) = 2 + 9 * 3 = 2 + 27 = 29
The solution :
x = - 3 , y = 29
Proof :
9 x + y = 2
9 * ( - 3 ) + 29 = 2
- 27 + 29 = 2
2 = 2
– 4 x – y = – 17
- 4 * ( - 3 ) - 29 = - 17
4 * 3 - 29 = - 17
12 - 29 = - 17
- 17 = - 17
To solve the system of equations:
Step 1: Choose one equation to solve for one variable (either x or y). Let's solve the first equation for y.
9x + y = 2
Subtract 9x from both sides:
y = 2 - 9x
Step 2: Substitute the expression for y into the second equation.
-4x - y = -17
Replace y with (2 - 9x):
-4x - (2 - 9x) = -17
Step 3: Simplify the equation.
-4x - 2 + 9x = -17
Combine like terms:
5x - 2 = -17
Step 4: Isolate the x-term by adding 2 to both sides.
5x = -17 + 2
5x = -15
Step 5: Divide both sides by 5 to solve for x.
x = -15/5
Simplify:
x = -3
Step 6: Substitute the value of x back into either of the original equations. Let's use the first equation.
9x + y = 2
Replace x with -3:
9(-3) + y = 2
-27 + y = 2
Step 7: Isolate the y-term by adding 27 to both sides.
y = 2 + 27
y = 29
Therefore, the solution to the system of equations is x = -3 and y = 29.
To solve this system of equations, you can use the method of elimination or substitution. I will explain both methods, and you can choose the one that you feel most comfortable with.
Method 1: Elimination
Step 1: Multiply one or both equations by a number(s) to make the coefficients of either x or y the same in both equations. In this case, we can multiply the second equation by -1 to make the coefficients of y the same.
-1(-4x - y) = -1(-17)
4x + y = 17
Step 2: Add the modified equations together, eliminating one variable.
(9x + y) + (4x + y) = 2 + 17
13x + 2y = 19
Step 3: Solve the resulting equation for one of the variables. In this case, let's solve for x.
13x = 19 - 2y
x = (19 - 2y) / 13
Step 4: Substitute the value of x back into one of the original equations and solve for the other variable. Let's substitute it into the first equation.
9((19 - 2y) / 13) + y = 2
Multiply both sides of the equation by 13 to eliminate the fraction.
9(19 - 2y) + 13y = 26
171 - 18y + 13y = 26
-5y = -145
y = -145 / -5
y = 29
Step 5: Substitute the value of y back into the equation used in Step 3 to solve for x.
x = (19 - 2(29)) / 13
x = (19 - 58) / 13
x = -39 / 13
x = -3
Therefore, the solution to the system of equations is x = -3 and y = 29.
Method 2: Substitution
Step 1: Start by solving one of the equations for one variable (either x or y). Let's solve the first equation for y.
y = 2 - 9x
Step 2: Substitute the expression for y in the second equation with the value obtained in Step 1.
-4x - (2 - 9x) = -17
Step 3: Simplify and solve the resulting equation for x.
-4x - 2 + 9x = -17
5x - 2 = -17
5x = -17 + 2
5x = -15
x = -15 / 5
x = -3
Step 4: Substitute the value of x back into one of the original equations and solve for the other variable. Let's substitute it into the first equation.
9(-3) + y = 2
-27 + y = 2
y = 29
Therefore, the solution to the system of equations is x = -3 and y = 29, which matches the solution obtained using the elimination method.