Question

how to solve equations with the substitution method

Answer 1

Answer:

In a system, the substitution method is one of the 3 main ways to solve a system and can be very efficient at times.

Skills needed: Systems, Algebra

Step-by-step explanation:

1) Let's say we are given two equations below:

  $y=4x \\ 4x-5y=-32$

  We can use substitution here by substituting in for $y$ in the second equation.   This means we put in $4x$ for $y$ in the 2nd equation so we only have $x$ variables in the equation, allowing us to solve for $x$.

2) Solving it out:

$1) 4x-5(4x)=-32 \\ 2) 4x-20x=-32 \\3) -16x=-32 \\4)x=2\\5)y=4x = 4(2) = 8 \\ 6)y=8$

We essentially substitute in that value as seen in step 1. Steps 2 and 3 are just simplifying the left side and allowing for us to solve. Step 4 is where we divide by -16 on both sides to solve for x. Step 5 and 6 show us solving for y using the value for x. We get $x=2,y=8$

Answer 2

Explanation:

The substitution property of equality says ...

  if A = B and A = C then B = C.

The value B can be put in place of A, because they are equal to each other.

This idea is used to solve equations by making a substitution that reduces the number of variables involved. This is done by writing an expression for one of the variables in terms of the other variable(s).

Example 1

In a 2-variable system of equations, for example, we might have ...

  y = 3x +2

  y = x -4

Each equation gives an expression for y in terms of x. That expression can be used where y is in the other equation.

  3x +2 = x -4 . . . . . using the expression for y of the first equation to substitute for y in the second equation.

This gives a 3-step equation in one variable, so is easily solved for the value of x. Then either of the original equations can be used to find the value of y. Here, x=-3, so y=-3-4 = -7 (using the second equation to find y).

Example 2

The idea of substitution can be used to solve equations of all kinds. Sometimes, the substitution is a bit messy. Nevertheless, it can reduce the problem to one that is more manageable.

A recent Brainly question involved the equations ...

  2y = 3x -1

  4x^2 -15xy +9y^2 = 0

This system reduces to a single quadratic in one variable by substituting for x or y in the second equation. We used y = (3x -1)/2, so the second equation became ...

  4x^2 -15x(3x -1)/2 +9((3x -1)/2)^2 = 0 . . . . the bold expression subs for y

This requires some simplification to get it to the usual quadratic form, but the problem can now be solved in a straightforward way.

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Additional comments

You can choose the substitution you want to make so that it simplifies your work. You might want to choose to substitute for a variable that has an easy expression to find. In a system of linear equations, you will usually look first for a variable with a coefficient of (plus or minus) 1. It will generally be easy to write the expression that can be substituted for that variable.

Sometimes, you don't need to find an expression for the "bare" variable. You may be able to substitute easily for some multiple of the variable.

Example 3

  2x +3y = 4

  3x +6y = 7

The value 6y is twice the value 3y, so we can write these equations as ...

  3y = 4 -2x . . . . . first equation solved for 3y

  2x +2(3y) = 7 . . . second equation written in terms of 3y

Now, we have an expression for 3y and a place to substitute it.

  2x +2(4 -2x) = 7 . . . . . 4 -2x substituted for 3y