When you're facing systems of equations, the SAT often provides you a shortcut if you recognize it. When a systems of equation question asks you to solve for a combination of variables (e.g.) and not just a single variable, there's often a faster way to solve directly for the combination.

Here, recognize that if you simply add the two equations together - much like using the "Elimination Method" but without actually trying to eliminate a variable - you can get一个dtogether with the same coefficient.

sums to:

If you then divide both sides by 8, you can get to exactly the answer they're looking for:

This system of equations provides you an excellent opportunity to use the Elimination Method to isolate a single variable. With the two equations provided, you already have a negativeterm (in the second equation) and a positiveterm (in the first). If you multiply the second equation by 2, you can get the coefficients the same and sum the equations to arrive at a single variable,. First, multiply the entire second equation by 2:

Then stack the updated equations and sum them:

Gives you:

This means thatbut remember to always double check that you've solved for the proper variable. The question wants to knowso you can plugback into either equation to solve. Using the original second equation:

So

这个问题问you to solve for, and conveniently provides you with a single-variable equation that allows you to solve directly for所以,你中途家里已经:

, so if you divide both sides byyou'll find that.

You can then plug into the first equation to get:

Addingto both sides gives you:

And then divide both sides byto get.

You now have your values for一个dso you can multiply them to get:

, makingthe correct answer.

This problem affords you a quick opportunity to use the Elimination Method. The first equation has a positiveterm and the second has a negativeterm, meaning that if you sum the two equations you will eliminate theterms and be left with a single variable,.

The two equations sum to:

And dividing both sides byallows you to determine that.

Note that the question asks for, not, so you need to plugback in to one of the two equations to solve for. Using the first equation, you have:

Someaning that.

One way to look at a system of linear equations is that the solution to that system is the point at which the graphs of the lines intersect at the samepoint. So a system of linear equations WILL NOT have any solutions if the lines never meet; in other words, if the lines are parallel with different y-intercepts.

The "by the book" method to determine if lines are parallel is to put each into slope-intercept form,, and then see if the slopesare the same. For the first line, that's:

So for the second line, you would need to find thevalue that makes the slope equal to -2. To get closer to slope-intercept form of the second equation you can start at:

And then divide both sides byto isolate theterm:

Ifyou've matched the slope of, makingthe correct answer.

Of course, there's a shortcut to this. If you recognize that between the two equations, the scale factor from first to second is thatis multiplied by 3, you can choose a value forthat provides the same effect for theterm. Sinceis multiplied byin the second equation, you can say that:

So.

For this problem, there are reasons to choose either the Elimination Method or the Substitution Method to solve the system. The question asks for the value of一个d the first equation gives you a great opportunity to substitute forin terms of一个d use the Substitution Method. Since, you can conclude that:

And then substitutewhere出现在第二个方程:

Distribute the multiplication across parentheses to get:

And then solve:

Of course, you could also use the Elimination Method. If you multiply the first equation by -3, you'd get:

, which you can stack with the second equation and sum:

When you sum, theterms cancel leaving you with the answer:

This system of equation gives you an opportunity to use the Elimination Method to quickly eliminate the variable一个d then use a single-variable equation to solve for. If you multiply the first equation by 3, you will then have aterm in the first equation and aterm in the second, so adding the two equations will eliminate the.

Once you've multiplied the first equation by 3, you'll sum the equations:

Adding these together gives you:

And then you can solve forby multiplying each side of the equation byto get.

Note that you should always double check that you've solved for the right variable (or combination of variables) for the question. This question does ask forso your answer is.

You can subtract the second equation from the first equation to eliminate:

You could also solve one equation for一个d substitute that value in forin the other equation:

Whenever you have a chance to solve a system of equations using the Elimination Method, it is generally your fastest option. Here if you multiply the first equation by 2, you can then subtract the two equations:

gives you a new first equation of. When you then stack and subtract:

You're left with:

So.

Then just plugback in to one of the two equations and you can solve for. If you use the first, you'll have:

So. This means that the ordered pair is.

This problem provides you with a quick opportunity to use the Elimination Method. Because there is anin the first equation andin the second, if you add the two equations you can eliminate theterms and solve straight for:

This means that.

Then you can pluginto either of the equations and you'll solve for. Using the first equation, that would be:

So

This means that the correct ordered pair is.