#Solve by substitution plus#

Here we go if I want to find x, I’m going to subtract 2 Xs from both sides, so now I have x plus 1 equals -3. I’m going to find x, and then go back and find my y value. Now this is a straight forward solving problem. That guy is equal to y and that guy is also equal to y, so I’m just substituting those two equations so they look like one equation with one variable. Since I know y is equal to the expression 3x plus 1, and y is also equal to the expression 2x minus 3, it makes sense mathematically to write 3x plus 1 equals 2x minus 3. We check our work by substituting those values into the other equation.This problem using substitution is going to be not too difficult because I have 2 equations that are both already solved for y. Okay, so now we know that #y=7# and #x=10#. Now we can find #x# by substituting in the #7# for #y#. We will leverage this information with the second equation by replacing the #x# with #17-y#. Now we see that #x# and the expression #17-y# are the same quantity.

Lets use the first equation and solve for #x#. First, pick one of equations and then solve for either #x# or #y#. We have to eliminate one of the variable by substitution. Neither one of these equation can be solved on their own because an equation with more than one unknown value cannot be solved. I think that this is better explained with an example. An easy way to check for parallel lines is to check if your two equations have the same slope, but different y intercepts. Therefore, they have no points in common, and therefore no common solutions. Lastly, if there are no solutions to a system, it means that the two lines represented by the equation are parallel.įor example, consider the following equations:Īs you can see, the two lines are parallel, and will never meet. Hence we can say that they share ever single solution that is valid for them, or rather have infinitely many solutions that work for the both of them. For example, if we took our graph from last time but made both lines #y=2x#:Īs you see, the two lines are indiscernible from one another. If there are infinitely many solutions, it means that both your equations are referring to the same line. This means that #(3,6)# is a solution to both equations, or in simpler terms, if you plugged in 3 for #x# in either equation, you would get 6 for #y#. If we graphed them both, this is what we'd end up with:Īs you can see, both lines intersect at the coordinate #(3,6)#. For example, considering the following equations: If there is one solution, it means that there is a single intersection between the two lines that your equations give.

With linear systems of equations, there are three possible outcomes in terms of number of solutions: Find the real values of #x# and #y# that satisfy the following systems of equations.Ĭ) #2x^2 - 3y^2 = -10, x^2 - 2x + 3y = 5#.Use the following practice exercises to develop your comfort with the skills dealt with in this answer. This will give you real roots of #2# and #-2#. It looks easiest to isolate the #y# in the first equation, however solving this equation won't be as neat as solving the previous one. Once again, as with the last example, we need to solve for one of the variables in one of the equations. This is found by inserting #x = 1# into one of the equations and solving for #y#.Įxample 2: Find all real values of #x# and #y# that satisfy the following system of equations: #3y = -2x^2 + 2, 2x^2 - 3y^2 = -4# We can now substitute into the other equation: I think it would be easiest to solve for #y# in the first equation. Since we want to solve with substitution, we must solve for one variable in one of the equations. So, I've prepared a couple of problems that I will work through slowly and carefully, showing all the steps to the final answer.Įxample 1: Solve the following system of equations- #2x + y = 5, 3x + 2y = 9# I'm not sure where exactly you mean 'final' in the solving process.

Solve by substitution