Solving Systems of Equations by Linear Combination (Elimination) Transcript

Question

NARRATOR: How do you solve systems of equations by using linear combination? For example, how do you solve the following system by using linear combination? Negative 3x plus 7y equals -16, and -9x plus 5y equals 16.

First equation reads: negative 3 times x plus 7 times y equals negative 16.

Second equation reads: negative 9 times x plus 5 times y equals 16.

NARRATOR: In this lesson, you will learn how to solve a system of equations by using linear combination, which is sometimes also called elimination.

So first, let's review.

If we have this system, -3x plus 7y equals -16, and -9x plus 5y is equal to 16. If we wanted to
solve this equation, we've already learned how to do it by substitution.

However, substituting for this system would get quite complicated.

Because if we were going to solve for one of these variables, they all have coefficients.
So that means there would be a lot of fractions involved. It is possible, but the math is going to get quite complicated.

So that's why we're going to talk about this way of elimination or linear combination. Because when all of your numbers or all of your variables have coefficients, this way is much simpler.

So first, let's review how to approximate the solution by graphing.

Text reads: Let’s Review

A vertical and a horizontal axis with the horizontal axis ranging from negative 10 to 10 in unit increments.

NARRATOR: So with these two equations, since they're in standard form, we're going to find their intercepts.

So for the green equation, the x-intercept is about 5 and 3/10. We can figure that out by putting in 0 for y and then solving for x.

The first equation, referred to here as the green equation, reads: negative 3 times x plus 7 times y equals negative 16.

Text reads: x-intercept is left parenthesis approximately symbol 5.3 comma 0 right parenthesis.

NARRATOR: And then the y-intercept of the green equation is about -2 and 3/10.

Text reads: y-intercept is left parenthesis 0 comma approximately symbol negative 2.3 right parenthesis.

NARRATOR: Again, we can solve that by substituting in 0 for x and solving for y.

Then we can use the same concept to find the x and the y-intercepts for the blue equation.

The second equation, referred to here as the blue equation, reads: negative 9 times x plus 5 times y equals 16.

Text reads: x-intercept is left parenthesis approximately symbol 1.8 comma 0 right parenthesis.

Text reads: y-intercept is left parenthesis 0 comma 3.2 right parenthesis.

NARRATOR: Now, we can plot to these points for both equations.

We're going to color code them and then connect them.

Two green points are plotted on the graph at left parenthesis 5.3 comma 0 right parenthesis and left parenthesis 0 comma negative 2.3 right parenthesis. A green line with arrows at both ends passes through the points. Values are approximate.

Two blue points are plotted at left parenthesis 1.8 comma 0 right parenthesis and left parenthesis 0 comma 3.2 right parenthesis. A blue line with arrows at both ends passes through the points. Values are approximate.

NARRATOR: And then, remember, each line here represents the set of all of the solutions to each of those equations.

So their intersection point is where both equations are true, and we can see here that
it's at about -4, -4.

Again, with graphing, especially ones like these where you have decimal answers, it's really hard to be exact. So that's why this guess, or this solution here, is just an approximate solution.

We're going to solve it algebraically by using linear combination to find the exact solution now, but we should keep in mind that it should be about -4, -4.

Intersection coordinate reads: left parenthesis approximate symbol negative 4 comma 4 right parenthesis.

Text reads: Core Lesson

NARRATOR: OK, now, to solve this by linear combination, what we need to do is when the two
equations get added together, we want one of our variables to sum to zero here.

On the left side of the screes a first equation reads: negative 3 times x plus 7 times y equals negative 16.

Second equation reads: negative 9 times x plus 5 times y equals 16.

NARRATOR: It's the easiest if you can find a variable that is a divisor or a multiple of that other corresponding variable in the other equation.

So we can see here that three is a divisor of nine.

When the x's add together, we want them to sum to 0.

So we need to multiply the first equation by -3 because then the coefficient of x will be positive 9, and when we add that to the second equation, they'll sum to 0 and cancel out.

First equation now reads: negative 3 times left parenthesis negative 3 times x plus 7 times y right parenthesis equals negative 3 times left parenthesis negative 16 right parenthesis.

NARRATOR: So if we multiply that out in the first one, we have 9x minus 21y equals positive 48, and then we'll just rewrite the second equation.

On the right side of the screen a new first equation reads: 9 times x minus 21 times y equals 48
New second equation reads: 9 times x plus 5 times y equals 16.

NARRATOR: Now, when we add these two equations together, 9x and negative 9x will cancel out or sum to 0.

On the right a new third equation sums the two equations to read: negative 16 times y equals 64.

NARRATOR: Just as a reminder, we are allowed to multiply an equation by a constant because of the multiplication property of equality that says, if you have an equation, you can multiply both sides by the same thing and still get a true equation.

We're allowed to add the two equations because of the addition property of equality.

Now, we have one equation with one variable, -16y equals 64.

So then, we multiply by negative 1/16 because that's the multiplicative inverse of -16, 83 and then we get that y is negative 4.

On the right side of the screen equation reads: negative 16 times y equals 64. Equation shows both sides of the equation being multiplied by negative one-sixteenth, for a solution that reads y equals negative 4.

NARRATOR: Now, in order to solve for x, we need to substitute that back into one of the original equations.

It doesn't matter which original equation you use. So I'm going to use the second equation here.

On the left side of the screen the second equation reads: negative 9 times x plus 5 times y equals 16.

NARRATOR: So if we substitute in -4 for y and simplify, then we can solve for our x and figure out what our x needs to be here.

Equation now reads: negative 9 times x plus 5 times left parenthesis negative 4 right parenthesis equals 16.

Next line reads: negative 9 times x minus 20 equals 16.

Next line reads: Both sides of the equation have positive 20 added to make a new equation negative 9 times x equals 36.

NARRATOR: So we use the additive inverse of -20, and then the multiplicative inverse of -9, and then we can see that x is equal to -4.

Equation reads: negative 9 times x equals 36.

Both sides of the equation are multiplied by negative one-ninth.

Solution reads: x equals negative 4.

NARRATOR: So our solution here is -4, -4.

Solution reads: left parenthesis negative 4 comma negative 4 right parenthesis.

NARRATOR: Remember when we graphed these two equations, their intersection point was about at that.

So that's a good check to make sure we had the right answer.

summarize

GPT-4o mini · Answer

The transcript explains how to solve a system of equations using linear combination (also known as elimination). The example provided features the following two equations:

\(-3x + 7y = -16\) (the green equation)
\(-9x + 5y = 16\) (the blue equation)

To solve these equations, the narrator highlights that using substitution would complicate the process due to the presence of coefficients. Instead, the linear combination method will simplify finding the solution.

Here's a step-by-step summary of the process:

Graphing the equations: The narrator briefly explains how to find the x- and y-intercepts for both equations and notes that the approximate intersection point from graphing is \((-4, -4)\).
Identifying coefficients: The narrator mentions that to eliminate one variable, one equation can be multiplied by a constant so that the coefficients of that variable in both equations become opposites.
Multiplying the first equation: The first equation is multiplied by -3, transforming it into: \[9x - 21y = 48\] The second equation remains the same: \[-9x + 5y = 16\]
Adding the equations: When both equations are added, the \(x\) terms cancel out: \[-16y = 64\]
Solving for \(y\): The equation is simplified, giving \(y = -4\).
Substituting back to find \(x\): The value of \(y\) is substituted back into one of the original equations (the second equation is used here): \[-9x + 5(-4) = 16\] Simplifying finds \(x = -4\).

The final solution to the system is \((-4, -4)\), which confirms the approximate intersection point noted earlier in the graphing step. The lesson demonstrates the efficiency of linear combination in solving systems of equations compared to substitution.