If the constants are not multiples, or do not have the same multiplier as the coefficients, then the system will have no solution. Using this knowledge, it is possible to recognize systems of linear equations in which there is no solution because the coefficients of each set of variables will be multiples of each other, and both pairs will have the same multiplier (the number you multiply to get the multiple). Therefore, there is no solution to the system of equations. Since the lines are parallel, they have the same slope, and will never intersect. The result obtained through the elimination method also yields a false statement since 0 ≠ 3 Using 2y and 4y results in a relatively easy equation to work with, since you can multiply the first equation by -2 to cancel out the 4y in the second equation, as shown below. You could also multiply the top equation by 5 and the bottom one by -6, but this also produces numbers that are more difficult to work with than the option presented below.
You could multiply the top equation by (or the bottom one by but this would make the rest of either equation more difficult to work with. In the example, 6x and 5x are more difficult to manipulate to cancel out than 2y and 4y. Ask yourself: "What can I multiply this variable by to cancel out the variable in the other equation?" To decide which equation to use and what constant to multiply it by, we can examine each pair of variables. To accomplish this, you need to multiply one, or sometimes both, of the equations in the system by a constant that will create a coefficient that is of equal value, but opposite sign from the corresponding variable in the other equation. This means that when you have the equations lined up by column and variable, as in the example above, you want the sum of the column of a variable to equal 0. The goal of elimination is to cancel out a variable. When using the elimination method, try to think of the system of equations as one big addition column, except that there is no carrying, and you can only add and subtract corresponding variables and constants. This organizes the systems of equations by aligning each term in one equation with their corresponding term in the other. Before starting, it is very important that both equations are in the standard form Ax + By = C. To solve systems of equations using the elimination method, you need to eliminate one variable. If both equations return true statements, as shown below, then you have successfully used the subtitution method! x = 10 You can check your answers by plugging the x- and y-values into the original equations and simplifying. Now you have the values of both coordinates! The solution to the system of equations is (10, 1). To solve for the remaining variable, you can plug in the value of the variable you just solved for into either of the original equations.
Now, because there is only one variable left in the equation, you can solve for that variable! Next, substitute the equivalent of the variable you just solved for into the corresponding variable in the other equation.
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