3. No Solution and Infinite SolutionsNo solution means parallel lines: same slope, different y-intercept. Infinitely many solutions means the same line: same slope and the same y-intercept. Get both sides into slope-intercept form, then match the slopes (and, for infinite, the intercepts) and solve for the constant.Worked example in the app →4. Mixture ProblemTwo cups going into one cup. Every cup gets a percent and an amount, and the equation is (percent)(amount) + (percent)(amount) = (percent)(amount). If one amount is x, the other one is the total minus x.Worked example in the app →
Systems of two linear equations are the part of Algebra where Desmos pays for itself. Two equations, two unknowns, and the answer is the point where the graphs cross. College Board knows you have the calculator, so the hard questions are built so that graphing is not enough: they ask for an expression instead of x, or they hide the numbers in a story, or they ask about a system with no solution at all.
The one move
Type both equations into Desmos and click the intersection. That is the solution, and it works no matter how the equations are written. Everything below is about what the question does after that point.
The five ways the SAT asks it
1. Solve it
Systems of linear equations · Mediumanswer D
The solution to the given system of equations is (x,y). What is the value of x?
Both equations in, click the point, x = 6. By hand you would add the equations and the 3y terms cancel, which is the quickest algebra when the same term appears with opposite signs. Either way the only trap is answering with y when they asked for x.
2. Set up the system from words
Systems of linear equations · Easyanswer A
A venue sells standard tickets for $5 each and premium tickets for $12 each. If 250 tickets are sold for a total of $2,300, which system of equations represents this situation, where s is the number of standard tickets and p is the number of premium tickets?
Two facts give two equations. Count equation: s + p = 250. Money equation: 5s + 12p = 2,300. The wrong choices put the total tickets with the prices or swap the $5 and $12. Match each number to its sentence and the system writes itself.
3. No solution, or infinitely many
Systems of linear equations · Mediumanswer D
Which of the following systems of linear equations has no solution?
Lines with the same slope and different intercepts never cross, so the system has no solution. Same slope and same intercept is one line, infinitely many solutions. Only one choice has two lines with matching slopes and different intercepts. Or type each pair into Desmos and see which one looks parallel. On the hard version they hand you one equation with a constant in it and ask what value makes the system have no solution: isolate y in both, set the slopes equal, solve for the constant.
4. They want an expression, not x
Systems of linear equations · Hardanswer B
If (x,y) is the solution to the given system of equations, what is the value of 5x+7y?
Desmos gives you the point. Then you still have to compute 5x + 7y, and the solution here has fractions, so keep the exact values on the calculator and type the expression in. Some of these are designed so the expression is easier to get than x and y: subtracting the equations gives 5x + 7y directly. If the numbers look ugly, look for that shortcut first.
5. The mixture problem
Systems of linear equations · Hardanswer A
A sample of a certain solution has a total mass of 60.0 grams and is 40.0% salt by mass. The sample was created by combining two portions of different solutions. The first portion was 25.0% salt by mass and the second portion was 70.0% salt by mass. What was the mass, in grams, of the second portion?
Two cups going into one cup. Every cup gets a percent and an amount, and the equation is percent times amount plus percent times amount equals percent times total. If one portion is x grams, the other is 60 − x. Type it into Desmos and read x. This one question shape is worth learning cold because it appears in three different skills on the test.
Try one
This is a real question from our Systems of Equations · Medium set. Solve it the fast way, then watch how I do it.
try one · Systems of equations · Medium
If the given system of equations has solution (x,y), what is the value of x+y?
3x+5y=72 5x+3y=56
Where the points actually go
Answering with the wrong variable, setting up the story backwards, and not recognizing the no-solution question for what it is. The calculator does the arithmetic. The reading is on you, and that only gets automatic through volume.
The app has 476 systems questions, 119 easy, 175 medium and 182 hard, every one with a video explanation. The free diagnostic shows you whether this topic is where your points are going.
Common Questions
Type both equations into the built-in calculator exactly as written and click where the two lines cross. The intersection point is the solution. It works for any form of the equations, with fractions and decimals included.
When the two lines are parallel: same slope, different y-intercept. Infinitely many solutions means the two equations describe the same line, same slope and same intercept. Isolate y in both equations and compare.
Each sentence with a total is one equation. A count total gives something like s + p = 250; a money or points total gives prices times counts, like 5s + 12p = 2,300. Match every number to the sentence it came from.
Two solutions with different percents combined into one. Set up percent times amount plus percent times amount equals percent times total, with one unknown amount as x and the other as the total minus x, then solve.
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