Solving a System of Equations by Substitution

Last updated about 3 years ago

14 questions

Note from the author:

Instructions

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Solving a System of Equations by Substitution

Last updated about 3 years ago

14 questions

Note from the author:

Step by step solution of a system of 2 linear equations by substitution.

Instructions

We're going to solve the following system of equations by substitution:

1)   -2x + y = 6
2)   3x - 2y = 4

We'll take it step by step.  Let's go!

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1) -2x + y = 6
2) 3x - 2y = 4

Let's solve equation 1 for y. What do you get when you add 2x to both sides of the equation?

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The next step is to _______ the expression we just found into the other equation, equation 2.  This is why this method is called solving by substitution.  

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When we take the step we talked about in question 5, equation 2 becomes

3x - 2(_______ ) = 4

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Now we have one equation with a single variable, x. The equation is 

3x - 2(2x+6) = 4 

Our next step is to solve this equation to figure out what the value of x will be in the solution to the system of equations.  To do that, we need to first get rid of the parentheses in our equation using the distributive property.  I've done part of this step in the equation below.  What goes in the blank space?

3x - _______ - 12 = 4

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So now we know what negative x equals, but we are looking for the value of x. To determine the value of x, we can now either multiply both sides of the equation by -1, or divide both sides by -1. When we do this, what do we get for the value of x?

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Now we need to figure out the value of the y coordinate in our solution. To do that, we will replace the x in one of our equations with the value we just got to get a single variable equation, which we then solve for y. Since we already solved one of the equations for y, it will be easiest to use that one, but you could also use either of the two original equations.
1) -2x + y = 6 or y = 2x + 6
2) 3x - 2y = 4

What single-variable equation do you get when you put in the known value of x?

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Simplify the equation you found in question 11. What is the value of y?

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If the answer to either question 13 or 14 is no, then you've made a mistake somewhere.  Either re-trace your steps to find the mistake, or start again.  Remember that the solution to a system of equations is the set of values for the variables that satisfy all of the equations in the system!

Step by step solution of a system of 2 linear equations by substitution.

We're going to solve the following system of equations by substitution:

1)   -2x + y = 6
2)   3x - 2y = 4

We'll take it step by step.  Let's go!

The first step is to solve one of the equations in the system for one of the variables. Which equation and variable do you think we should choose?

Solve equation 1 for x

Solve equation 1 for y

Solve equation 2 for x

Solve equation2 for y

1) -2x + y = 6
2) 3x - 2y = 4

Let's solve equation 1 for y. We can do this one of two ways. Which of the following will work to do that (select two)?

Add -2x to both sides of the equation

Subtract -2x from both sides of the equation

Add 2x to both sides of the equation

Subtract 2x from both sides of the equation.

What property allows us to add the same number to both sides of an equation?

Associative property of addition

Distributive property

Addition property of equality

Reflexive property

1) -2x + y = 6
2) 3x - 2y = 4

Let's solve equation 1 for y. What do you get when you add 2x to both sides of the equation?

The next step is to _______ the expression we just found into the other equation, equation 2.  This is why this method is called solving by substitution.  

When we take the step we talked about in question 5, equation 2 becomes

3x - 2(_______ ) = 4

Now we have one equation with a single variable, x. The equation is 

3x - 2(2x+6) = 4 

Our next step is to solve this equation to figure out what the value of x will be in the solution to the system of equations.  To do that, we need to first get rid of the parentheses in our equation using the distributive property.  I've done part of this step in the equation below.  What goes in the blank space?

3x - _______ - 12 = 4

Next, we need to combine like terms on both sides of the equation. For this problem, we only have like terms on the left side, 3x and -4x. What do we get when we combine them?

Our equation now looks like this:

-x - 12 = 4

Now we want to move all the constant terms to the right side of the equals sign. How do we do this (select two)?

Add -12 to both sides

Add 12 to both sides

Subtract 12 from both sides

Subtract -12 from both sides

So now we know what negative x equals, but we are looking for the value of x. To determine the value of x, we can now either multiply both sides of the equation by -1, or divide both sides by -1. When we do this, what do we get for the value of x?

Now we need to figure out the value of the y coordinate in our solution. To do that, we will replace the x in one of our equations with the value we just got to get a single variable equation, which we then solve for y. Since we already solved one of the equations for y, it will be easiest to use that one, but you could also use either of the two original equations.
1) -2x + y = 6 or y = 2x + 6
2) 3x - 2y = 4

What single-variable equation do you get when you put in the known value of x?

Simplify the equation you found in question 11. What is the value of y?

Put the value you found for x and y into the original equation 1. Do you get a true statement?

Yes

No

Put the value you found for x and y into the original equation 2. Do you get a true statement?

Yes

No

If the answer to either question 13 or 14 is no, then you've made a mistake somewhere.  Either re-trace your steps to find the mistake, or start again.  Remember that the solution to a system of equations is the set of values for the variables that satisfy all of the equations in the system!

The first step is to solve one of the equations in the system for one of the variables.  Which equation and variable do you think we should choose?

Solve equation 1 for x

Solve equation 1 for y

Solve equation 2 for x

Solve equation2 for y

1)   -2x + y = 6
2)   3x - 2y = 4

Let's solve equation 1 for y.  We can do this one of two ways.  Which of the following will work to do that (select two)?

Add -2x to both sides of the equation

Subtract -2x from both sides of the equation

Add 2x to both sides of the equation

Subtract 2x from both sides of the equation.

What property allows us to add the same number to both sides of an equation?

Associative property of addition

Distributive property

Addition property of equality

Reflexive property

Next, we need to combine like terms on both sides of the equation.  For this problem, we only have like terms on the left side, 3x and -4x.  What do we get when we combine them?

7x

x

-12x

-x

12x

Our equation now looks like this:

-x - 12 = 4

Now we want to move all the constant terms to the right side of the equals sign.  How do we do this (select two)?

Add -12 to both sides

Add 12 to both sides

Subtract 12 from both sides

Subtract -12 from both sides

Put the value you found for x and y into the original equation 1.  Do you get a true statement?

Yes

No

Put the value you found for x and y into the original equation 2.  Do you get a true statement?

Yes

No

Solving a System of Equations by Substitution

Solving a System of Equations by Substitution

1) -2x + y = 62) 3x - 2y = 4Let's solve equation 1 for y. What do you get when you add 2x to both sides of the equation?

So now we know what negative x equals, but we are looking for the value of x. To determine the value of x, we can now either multiply both sides of the equation by -1, or divide both sides by -1. When we do this, what do we get for the value of x?

Simplify the equation you found in question 11. What is the value of y?

The first step is to solve one of the equations in the system for one of the variables. Which equation and variable do you think we should choose?

1) -2x + y = 62) 3x - 2y = 4Let's solve equation 1 for y. We can do this one of two ways. Which of the following will work to do that (select two)?

What property allows us to add the same number to both sides of an equation?

1) -2x + y = 62) 3x - 2y = 4Let's solve equation 1 for y. What do you get when you add 2x to both sides of the equation?

Next, we need to combine like terms on both sides of the equation. For this problem, we only have like terms on the left side, 3x and -4x. What do we get when we combine them?

Our equation now looks like this:-x - 12 = 4Now we want to move all the constant terms to the right side of the equals sign. How do we do this (select two)?

So now we know what negative x equals, but we are looking for the value of x. To determine the value of x, we can now either multiply both sides of the equation by -1, or divide both sides by -1. When we do this, what do we get for the value of x?

Simplify the equation you found in question 11. What is the value of y?

Put the value you found for x and y into the original equation 1. Do you get a true statement?

Put the value you found for x and y into the original equation 2. Do you get a true statement?

1) -2x + y = 6
2) 3x - 2y = 4

Let's solve equation 1 for y. What do you get when you add 2x to both sides of the equation?

1) -2x + y = 6
2) 3x - 2y = 4

Let's solve equation 1 for y. We can do this one of two ways. Which of the following will work to do that (select two)?

1) -2x + y = 6
2) 3x - 2y = 4

Let's solve equation 1 for y. What do you get when you add 2x to both sides of the equation?

Our equation now looks like this:

-x - 12 = 4

Now we want to move all the constant terms to the right side of the equals sign. How do we do this (select two)?