Number Problem The sum of two numbers is twice their difference. The larger number is 6 more tha

Number Problem The sum of two numbers is twice their...

Key Concepts

Translating Word Problems into Algebraic Equations

This concept involves reading a verbal statement and converting the relationships and conditions described into algebraic equations. It requires identifying the unknown quantities, defining variables, and setting up equations that represent the relationships among these variables. This translation is a fundamental step in connecting real-world problems to mathematical analysis.

Systems of Linear Equations

A system of linear equations consists of two or more linear equations involving the same set of variables. Solving such a system means finding values for the variables that satisfy all the equations simultaneously. This concept is crucial in many areas of mathematics and applied sciences, as it allows for the modeling of multiple constraints and interactions between quantities.

Algebraic Methods for Solving Equations

Once the equations are set up, algebraic methods such as substitution or elimination are used to find the values of the unknowns. Substitution involves solving one equation for one variable and then plugging that expression into the other equation(s), while elimination involves adding or subtracting equations to eliminate one of the variables. Mastery of these methods is essential for efficiently solving systems of linear equations.

Transcript

00:01 Okay, so we have a word problem and it states that the sum of two numbers, so we have two numbers is twice their difference.

00:11 Okay, so we have two numbers.

00:12 Let's label them.

00:14 So let's have x is equal to a number and y is equal to another number.

00:28 Okay, so it says the sum of two numbers, well, that's just x plus y addition, is twice is, which is equal to twice their difference.

00:43 So the difference is x minus y.

00:48 And now it says the larger number, a larger number.

00:54 Okay, so let's make one of the numbers larger than the other.

00:58 So let's say x is greater than y.

01:03 So that when we have x minus y, it would still be greater than zero.

01:07 So we have so larger number, x is equal to six more six more means addition to six plus then twice the smaller number which is y now we have to find the numbers okay so we wrote out our system now let's solve it okay since we have already x isolated why don't we plug it into the first equation so we get six plus 2y plus y is equal to 2x minus 2y expanded here so combining like the terms with 3y and wait i still have to plug in x into here okay let me rewrite this i need more space so i have 6 plus 2 y plus y is equal to 2x minus 2 y but 2 x is 2 x is 2 2 times 6 plus 2y minus 2y.

02:23 Okay, so now combining the length terms, i get 3y plus 6 as equal to 12 plus 4y minus 2, y, and that this gives me 2y...

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