SAT Math: systems of equations, the fast way

Hi! Systems of equations show up again and again on SAT Math. In this lesson you'll learn two solid methods, and one shortcut that saves you a full minute when the question is phrased the right way.

Your three tools

You have three tools. Elimination: add or subtract the equations so one variable disappears. Substitution: solve one equation for a variable and plug it into the other. And the shortcut: solve directly for what the question asks, not always for x and y.

• Elimination: Add or subtract the equations
• Substitution: Plug one into the other
• Shortcut: Solve for what they ask

Elimination

Here's elimination. x plus y equals ten, and x minus y equals four. Look: plus y and minus y cancel out if we add the equations. Adding gives two x equals fourteen, so x is seven. Put seven back into the first equation: seven plus y is ten, so y is three.

Substitution

Substitution is best when one equation already says y equals something. Here y equals two x plus one. So we replace y in the second equation with two x plus one. Three x plus two x plus one equals sixteen. Five x is fifteen, so x is three. Then y is two times three plus one, which is seven.

What a solution means

Here's what that answer means. Each equation is a line, and the solution is where they cross: at three, seven. On the digital SAT you have a built-in graphing calculator, so you can always check an answer this way.

The shortcut

Now the shortcut. The question asks for x plus y, not x and y separately. Look at the coefficients: three and two, then two and three. Add the equations: five x plus five y equals thirty. Divide by five: x plus y is six. Done. You never had to find x or y.

Your turn

Your turn. Two x plus y equals eleven, and x minus y equals one. What is x?

Same slope → no solution

One more classic. If two lines have the same slope but different intercepts, they're parallel. They never cross, so the system has no solution. The SAT loves to ask: for what value of k is there no solution?

The classic k question

Try the classic. For what value of k does this system have no solution?

Find the mistake

This student used the right method but slipped at the end. Tap the wrong line.

Fill the gaps

Quick recap. Fill in the gaps.

You can now

Nice work! You can now solve systems by elimination and substitution, answer what is x plus y in a single step, and find the value of k that makes a system have no solution.

• Solve by elimination and substitution
• Answer "what is x + y" in one step
• Spot the no-solution k