True
2x + 6y = 93x - 12y = 15Chose the second equation because you can simplify it.3x - 12y = 15 divide by 3 to both sidesx - 4y = 5 add 4y to both sidesx = 4y + 5Replace 4y + 5 for x to the first equation.2x + 6y = 92(4y + 5) + 6y = 98y + 10 + 6y = 914y = -1y = -1/14x = 4y + 5x = 4(-1/14) + 5x = -2/7 + 5x = -2/7 + 35/7x = 33/7
There must be fewer independent equation than there are variables. An equation in not independent if it is a linear combination of the others.
-1
This can happen in different ways: a) More variables than equations. For instance, a single equation with two variables (such as x + y = 15), two equations with three variables, two equations with four variables, etc. b) To of the equations describe the same line, plane, or hyper-plane - this, in turn, will result in that you "really" have less equations than it seems. For example: y = 2x + 3 2y = 4x + 6 The second equation is simply the first equation multiplied by 2.
First get y in terms of x: 5x-2y=-12 ----- -2y=-12-5x ------ y=6+(5/2)x Sub into other equation: 2x+3y=-1 ---- 2x+3(6+(5/2)x)=-1 Solving for x gets: x= -2 Sub into other equation: 5x-2y= -12 --- 5(-2)-2y= -12 --- y=1 So x= -2, and y= 1
The first step is to solve one of the equations for one of the variables. This is then substituted into the other equation or equations.
You use substitution when you can solve for one variable in terms of the others. By substituting, you remove one variable from the equation, which can then be solved. Once you solve for one variable, you can use substitution to find the other.
If you mean x+2y = -2 and 3x+4y = 6 then by solving the simultaneous equations by substitution x = 10 and y = -6
By elimination and substitution
The first step is usually to solve one of the equations for one of the variables.Once you have done this, you can replace the right side of this equation for the variable, in one of the other equations.
Substitution is often used when one of the equations in a system is already solved for one variable, or can be easily manipulated to do so. For example, if you have the equations (y = 2x + 3) and (3x + 2y = 12), substituting the expression for (y) from the first equation into the second allows for straightforward solving. This method is particularly useful when dealing with linear equations, as it simplifies the process of finding the variable values.
The first step is to show the equations which have not been shown.
Isolating a variable in one of the equations.
You'd need another equation to sub in
To solve this system of equations using substitution, we can isolate one variable in one equation and substitute it into the other equation. From the second equation, we can express x in terms of y as x = 4 + 2y. Then, substitute this expression for x into the first equation: 4(4 + 2y) - 3y = 1. Simplify this equation to solve for y. Once you find the value of y, substitute it back into x = 4 + 2y to find the corresponding value of x.
To solve this system of equations, we can use the method of substitution or elimination. Let's use the substitution method. From the second equation, we can express y as y = 55 - 4x. Substitute this expression for y in the first equation: 7x - 5(55 - 4x) = 76. Simplify this equation to solve for x. Then, substitute the value of x back into one of the original equations to find the value of y.
In the context of solving a system of equations by substitution, a "useless result" like 12 equals 12 indicates that the two equations are actually dependent, meaning they represent the same line or have infinitely many solutions. Instead of finding a unique solution, you end up with a tautology that confirms the equations are equivalent. This suggests that any solution that satisfies one equation will also satisfy the other, leading to an infinite set of solutions rather than a single point of intersection.