- 16-Jul-2022

**To Solve a System of Equations by Elimination**

- Write both equations in standard form.
- Make the coefficients of one variable opposites.
- Add the equations resulting from Step 2 to eliminate one variable.
- Solve for the remaining variable.
- Substitute the solution from Step 4 into one of the original equations.

It tells us that **if a quantity a equals quantity b, and b equals the quantity, c, then a and c are equal as well**. Since we know that 30 + 30 = 20 + 40 and that 30 + 30 = 60 we can substitute 30 + 30 for 20 + 40 and get 60 = 20 + 40. This is called the substitution property of equality.

The substitution method is **the algebraic method to solve simultaneous linear equations**. As the word says, in this method, the value of one variable from one equation is substituted in the other equation.

**Here's how it goes:**

- Step 1: Solve one of the equations for one of the variables. Let's solve the first equation for y:
- Step 2: Substitute that equation into the other equation, and solve for x.
- Step 3: Substitute x = 4 x = 4 x=4 into one of the original equations, and solve for y.

The method of solving "by substitution" works by solving one of the equations (you choose which one) for one of the variables (you choose which one), and then plugging this back into the other equation, "substituting" for the chosen variable and solving for the other. Then you back-solve for the first variable.

Substitution is **the name given to the process of swapping an algebraic letter for its value**. Consider the expression 8 + 4. This can take on a range of values depending on what number actually is. If we are told = 5, we can work out the value of the expression by swapping the for the number 5.

The method of substitution involves three steps: **Solve one equation for one of the variables**. Substitute this expression into the other equation and solve. Resubstitute the value into the original equation to find the corresponding variable.

To substitute a number into an algebraic expression, all you need to do is **re-write the expression in exactly the same way, except replacing the variable (letter) with the number**. It always makes it clearer to put the number in brackets too. Then you can simplify your new expression and you have your answer!

Substituting for a Variable**If you know the value that the variable is equal to, you can substitute that value in for the variable in the expression**! Let's look at an example. Image by Caroline Kulczycky. Since we know that x=5, we can directly substitute or replace the x in the expression x+3 with a 5 to solve for y!

**To Solve a System of Equations by Elimination**

- Write both equations in standard form.
- Make the coefficients of one variable opposites.
- Add the equations resulting from Step 2 to eliminate one variable.
- Solve for the remaining variable.
- Substitute the solution from Step 4 into one of the original equations.

The substitution property of equality makes it easy to verify any solution. Simply **substitute the solution back into the original equation anywhere the variable appears**. Then, simplify to ensure that the two sides are still the same.

Step 1 : In the given two equations, solve one of the equations either for x or y. Step 2 : Substitute the result of step 1 into other equation and solve for the second variable.

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