a chemical company makes two brands of antifreeze the first brand is 20% pure antifreeze and the second brand is 45% pure antifreeze in order to obtain 70 gallons of the mixture that contains 30% antifreeze how many gallons of each brand of antifreeze must be used

first brand: 42 gallons

second brand: 28 gallons

Step-by-step explanation:

So you know that the first brand is 20% antifreeze, this means that if you have x gallons then 0.20x is pure antifreeze. The same thing can be applied for the second brand except the coefficient is 0.45 and I'll just say the variable is y. If you haven't noticed, the coefficients are simply the percentages of antifreeze, but in decimal form. Anyways, since it's asking for a mixture that's 30% antifreeze and is 70 gallons that means 0.30(70) gallons is pure antifreeze. This simplifies to 21 gallons. So if we take the pure antifreeze of both mixtures we get the following equation:

the x represents how many gallons of the first brand, and the y represents how many gallons of the second brand. Since that's what it represents we also get the equation:

In this equation there aren't any coefficients, since x and y represent the whole amount. the 0.2x and 0.45 simply represent how much is pure antifreeze.

So to solve this systems of equation we can solve for y or x, either works, and then substitute it into the other equation. So in this example I'll solve for y

Original equation:

Subtract x from both sides

So now we can substitute this into the pure antifreeze question

So now we can solve for x since it's the only variable in this equation

Simplify

Subtract 31.5 from both sides

Now divide both sides by -0.25

So this means we have to use 42 gallons of the first brand, now we can substitute this into either equation to solve for y, but it's much easier to use the total gallon equation:

Subtract 42 from both sides

So this means we need to use 42 gallons of the first brand and 28 gallons of the second brand

answer: domain and range mean all possible values of x and y that could be substituted to the inequality. a solution means all possible values that make the inequality statement true. so while the range is still all real numbers, the solution is the shaded area, for which y will always be greater than the line of the function.