There are infinite solutions to the system. Determine the number of solutions. 1) y = – 3x + 4 y = – 3x – 1 There is no solution to the system. 2) 4x + 2y = 6 6x + 3y = 9 Rewrite the 1st equation in slope intercept form. Rewrite the 2nd equation in slope intercept form. 4x + 2y = 6 → 2y = – 4x + 6 → y = – 2x + 3 6x + 3y = 9 → 3y = – 6x + 9 → y = – 2x + 3 Both equations are in slope intercept form and the slopes are the same and the y intercepts are the same. y = – 2x + 3 There are infinite solutions to the system.
10.02 Solving Systems of Equations by Substitution
A system of equations can also be solved without graphing A system of equations can also be solved without graphing. One method that can be used is called the substitution method. To use the substitution method, rewrite the system with two variables as a system with one variable. To do this, substitute (replace) an equivalent expression for one of the variables. The equivalent expression can be found from one of the two equations in the system. Solve the equation that now has only one variable ( x or y ) and find the value for that variable.
Use the value of that variable ( x or y ) to find the value of the other variable. To do this, replace the variable with the solved value in either equation of the system and solve for the remaining variable. The x and y values ( x , y ) are the solution to the system. If the variables cancel out, then the remaining statement will either be true or false. If the statement is true, then there are infinite solutions. If the statement is false, then there is no solution.
Solve the following system of equations using the substitution method. y = 4x x + y = 10 Notice that the first equation is solve for y. Replace the y with 4x in the 2nd equation. x + (4x) = 10 Solve the equation for x. 5x = 10 → x = 2 Replace x with 2 in either equation from the system. y = 4x → y = 4(2) → y = 8 The solution to the system is ( 2 , 8 ) x = 2 and y = 8
Solve the following system using the substitution method. x = 2y + 1 x + 3y = 16 Notice that the first equation is solve for x. Replace the x with (2y + 1) in the 2nd equation. (2y + 1) + 3y = 16 Solve the equation for y. 5y + 1 = 16 → 5y = 15 → y = 3 Replace y with 3 in either equation from the system. x = 2y + 1 → x = 2(3) + 1 → x = 7 The solution to the system is ( 7 , 3 ) x = 7 and y = 3
Solve the following system using the substitution method. x = 5y + 6 3x – 2y = 5 Notice that the first equation is solve for x. Replace the x with (5y + 6) in the 2nd equation. 3(5y + 6) – 2y = 5 Solve the equation for y. 15y + 18 – 2y = 5 → 13y + 18 = 5 → 13y = – 13 → y = – 1 Replace y with – 1 in either equation from the system. x = 5y + 6 → x = 5(– 1) + 6 → x = 1 The solution to the system is ( 1 , – 1 ) x = 1 and y = – 1
Solve the following system using the substitution method. 2x + y = 9 y = – 2x + 7 Notice that the second equation is solve for y. Replace the y with (– 2x + 7) in the 1st equation. 2x + (– 2x + 7) = 9 Solve the equation for x. 2x – 2x + 7 = 9 → 0x + 7 = 9 → 7 = 9 There is no variable left and 7 = 9 is false. There is no solution to the system.
Solve the following system using the substitution method. 3x + y = 4 y = – 3x + 4 Notice that the second equation is solve for y. Replace the y with (– 3x + 4) in the 1st equation. 3x + (– 3x + 4) = 4 Solve the equation for x. 3x – 3x + 4 = 4 → 0x + 4 = 4 → 4 = 4 There is no variable left and 4 = 4 is true. There are infinite solutions to the system.
Solve the following system using the substitution method. Try This: Solve the following system using the substitution method. y = 3x + 1 4x + y = 8 Notice that the first equation is solve for y. Replace the y with (3x + 1) in the 2nd equation. 4x + (3x + 1) = 8 Solve the equation for x. 7x + 1 = 8 → 7x = 7 → x = 1 Replace the x with 1 in either equation from the system. y = 3x + 1 → y = 3(1) + 1 → y = 4 The solution to the system is ( 1 , 4 ). x = 1 and y = 4
Solve the following system using the substitution method. y = x – 10 x + y = 2 Notice that the first equation is solve for y. Replace the y with (x – 10) in the 2nd equation. x + (x – 10) = 2 Solve the equation for x. 2x – 10 = 2 → 2x = 12 → x = 6 Replace x with 6 in either equation from the system. y = x – 10 → y = 6 – 10 → y = – 4 The solution to the system is ( 6 , – 4 ) x = 6 and y = – 4