Question Video: Solving Simultaneous Equations by Elimination | Nagwa Question Video: Solving Simultaneous Equations by Elimination | Nagwa

Question Video: Solving Simultaneous Equations by Elimination Mathematics • Third Year of Preparatory School

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By first multiplying the equations to make the coefficients of either π‘₯ or 𝑦 equal, use the elimination method to find π‘₯ and 𝑦 by solving the simultaneous equations 2π‘₯ + 3𝑦 = 4, 3π‘₯ + 4𝑦 = 8.

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Video Transcript

By first multiplying the equations to make the coefficients of either π‘₯ or 𝑦 equal, use the elimination method to find π‘₯ and 𝑦 by solving the simultaneous equations two π‘₯ plus three 𝑦 equals four, three π‘₯ plus four 𝑦 equals eight.

We’ve been given specific instructions on the method to use to solve these simultaneous equations. We want to use the elimination method. In the elimination method, you either add or subtract to the equations to get an equation in one variable. This method works when the coefficients of one of the variables are equal to each other.

Because none of our coefficients are equal to each other, we’ll first multiply the equations by some value to make either the π‘₯ or 𝑦 equal: equation one two π‘₯ plus three 𝑦 equals four, and equation two is three π‘₯ plus four 𝑦 equals eight. The π‘₯-terms have coefficients of two and three, and the 𝑦-terms have coefficients three and four.

We could multiply through by many different values. But it’s simplest to look for the least common multiple. For example, with two and three as the π‘₯-coefficients, the least common multiple would be six. And to have a coefficient of six, we multiply the first equation through by three. This gives an equivalent equation: six π‘₯ plus nine 𝑦 equals 12. For our second equation, we multiply through by two, which produces six π‘₯ plus eight 𝑦 equals 16.

Using the elimination method, we can subtract the second equation from the first equation, being careful to remember that we are subtracting every term. So it might be helpful to distribute that subtraction across the equation. Six π‘₯ minus six π‘₯ equals zero, nine 𝑦 minus eight 𝑦 equals one 𝑦, and 12 minus 16 equals negative four.

Now that we found that 𝑦 equals negative four, we can plug 𝑦 back into our first equation, our second equation, or even these equivalent equations to solve for π‘₯. Using our first equation, two π‘₯ plus three 𝑦 equals four, if we plug in negative four for 𝑦, we see two π‘₯ minus 12 equals four. Adding 12 to both sides gives us two π‘₯ equals 16. And dividing both sides by two, we find that π‘₯ equals eight. The solution to this system of equations is π‘₯ equals eight and 𝑦 equals negative four.

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