Module 11.4

Solving Linear SystemsBy Multiplying First

How can you solve a system of linear equations by using multiplication and elimination?

P. 515

What do you do if all 4 variables have coefficients, and none can be eliminated?For example:

𝟑𝒙 + 𝟖𝒚 = 𝟕𝟐𝒙 − 𝟐𝒚 = −𝟏𝟎

The first term of both equations, 3x and 2x, can’t be added together, even if you multiply one of them by –1.The second term of both equations, 8y and –2y, can’t be added together.

But – we can multiply ONE ENTIRE equation by a number, which will make one of the coefficients become an opposite.

Specifically: If we multiply the whole 2nd equation by 4, here’s what it becomes:

𝟒 𝟐𝒙 − 𝟐𝒚 = −𝟏𝟎 ⇒ 𝟖𝒙 − 𝟖𝒚 = −𝟒𝟎

Did you notice that the new 2nd term, −𝟖𝒚, is the opposite of the 2nd term of the 1st equation, 𝟖𝒚 ?Here’s how they line up:

𝟑𝒙 + 𝟖𝒚 = 𝟕𝟖𝒙 − 𝟖𝒚 = −𝟒𝟎Now you can add the 2nd terms!

P. 518

P. 519Solving Systems of Equations by Multiplying First

Step 1 Decide which variable to eliminate.

Step 2 Multiply one or both equations by a constant so that adding the equations will eliminate the variable.

Step 3 Solve the system using the elimination method.

What do you do if all 4 variables have coefficients, and multiplying one whole equation doesn’t help?For example:

−𝟑𝒙 + 𝟐𝒚 = 𝟒𝟒𝒙 − 𝟏𝟑𝒚 = 𝟓

Multiplying the 1st term of the 1st equation by anything won’t give you –4.Multiplying the 1st term of the 2nd equation by anything won’t give you 3.Multiplying the 2nd term of the 1st equation by anything won’t give you 13.Multiplying the 2nd term of the 2nd equation by anything won’t give you –2.

But – we can multiply BOTH ENTIRE equations – each by a different number, which will make two of the coefficients become opposites.

For example: If we multiply the whole 1st equation by 4, here’s what it becomes:

𝟒 −𝟑𝒙 + 𝟐𝒚 = 𝟒 ⇒ −𝟏𝟐𝒙 + 𝟖𝒚 = 𝟏𝟔

And if we multiply the whole 2nd equation by 3, here’s what it becomes:

𝟑 𝟒𝒙 − 𝟏𝟑𝒚 = 𝟓 ⇒ 𝟏𝟐𝒙 − 𝟑𝟗𝒚 = 𝟏𝟓

Did you notice that both new 1st terms are opposites?Here’s how they line up:

−𝟏𝟐𝒙 + 𝟖𝒚 = 𝟏𝟔𝟏𝟐𝒙 − 𝟑𝟗𝒚 = 𝟏𝟓

Now you can add the 1st terms!

P. 519

P. 519Solving Systems of Equations by Multiplying FirstStep 1 Decide which variable to eliminate.

Step 2 Multiply one or both equations by a constant so that adding the equations will eliminate the variable.

Step 3 Solve the system using the elimination method.

Solving Systems of Equations by Multiplying FirstStep 1 Decide which variable to eliminate.Step 2 Multiply one or both equations by a constant so that adding the equations will eliminate the variable.Step 3 Solve the system using the elimination method.

P. 520

P. 520

Solving Systems of Equations by Multiplying FirstStep 1 Decide which variable to eliminate.

Step 2 Multiply one or both equations by a constant so that adding the equations will eliminate the variable.

Step 3 Solve the system using the elimination method.

P. 520