Math

SAT Suite of Assessments Skills Insight Tool

Below are the skills and knowledge that students in the content domain and performance score band selected above are typically able to demonstrate as well as examples of the kinds of questions that these students are likely able to answer correctly. To view skill/knowledge statements and example questions in other domains and/or performance score bands, update the selections above and click Go.

Skills

A student in this performance score band can typically demonstrate the following skills in this content domain:

With or without a context, create one linear equation or a system of two linear equations in two variables that models a situation, find and use the solution to a given system of linear equations, or determine the conditions under which a linear equation or system of linear equations has zero, exactly one, or infinitely many solutions
With or without a context, create a linear equation or inequality in two variables when given two input-output pairs, a table of values, features of parallel or perpendicular lines, or details about a translation of a given function

Example Questions

Example Question 1

In the xy-plane, line $k$ passes through the points $left parenthesis 8 comma 6 right parenthesis$ and $left parenthesis 9 comma 12 right parenthesis$ and is defined by the equation $y = m x + b$ . What is the value of $b$ ?

Key: -42

Key Explanation

The correct answer is $negative 42$ . For a linear equation in the form $y = m x + b$ , $m$ represents the slope and $b$ represents the y-coordinate of the y-intercept of the graph in the xy-plane. The slope of the graph of a line containing any two points, $left parenthesis x 1 comma y 1 right parenthesis$ and $left parenthesis x 2 comma y 2 right parenthesis$ , can be found using the slope formula $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ . It’s given that in the xy-plane, line $k$ passes through the points $left parenthesis 8 comma 6 right parenthesis$ and $left parenthesis 9 comma 12 right parenthesis$ . Substituting $left parenthesis 8 comma 6 right parenthesis$ and $left parenthesis 9 comma 12 right parenthesis$ for $left parenthesis x 1 comma y 1 right parenthesis$ and $left parenthesis x 2 comma y 2 right parenthesis$ , respectively, in the slope formula yields $m = \frac{12 - 6}{9 - 8}$ , or $m equals 6$ . Since line $k$ passes through the point $left parenthesis 9 comma 12 right parenthesis$ and the slope of the line is $6$ , substituting $9$ for $x$ , $12$ for $y$ , and $6$ for $m$ in the equation $y = m x + b$ yields $12 = (6) (9) + b$ , or $12 = 54 + b$ . Subtracting $54$ from both sides of this equation yields $- 42 = b$ . Therefore, the value of $b$ is $negative 42$ .

Example Question 2

$y - 6 = \frac{4}{5} (x + 5)$

$y - v = \frac{4}{5} (x + 10)$

In the given system of equations, $v$ is a constant. The system has infinitely many solutions. What is the value of $v$ ?

$0$
$2$
$4$
$12$

Key: B

Key Explanation

Choice B is correct. It's given that the system has infinitely many solutions. A system of two linear equations has infinitely many solutions when the two linear equations are equivalent. Adding $6$ to both sides of $y - 6 = \frac{4}{5} (x + 5)$ yields $y = \frac{4}{5} (x + 5) + 6$ . Applying the distributive property to this equation yields $y = \frac{4}{5} x + \frac{20}{5} + 6$ , or $y = \frac{4}{5} x + 10$ . Adding $v$ to both sides of $y - v = \frac{4}{5} (x + 10)$ yields $y = \frac{4}{5} (x + 10) + v$ . Applying the distributive property to this equation yields $y = \frac{4}{5} x + \frac{40}{5} + v$ , or $y = \frac{4}{5} x + 8 + v$ . For the equations $y = \frac{4}{5} x + 10$ and $y = \frac{4}{5} x + 8 + v$ to be equivalent, it follows that $10 = 8 + v$ . Subtracting $8$ from both sides of this equation yields $2 equals v$ . Therefore, the value of $v$ is $2$ .

Distractor Explanations

Choice A is incorrect. If $v equals 0$ , the system of equations would have no solution, not infinitely many solutions.

Choice C is incorrect. If $v equals 4$ , the system of equations would have no solution, not infinitely many solutions.

Choice D is incorrect. If $v equals 12$ , the system of equations would have no solution, not infinitely many solutions.

Reading and Writing

Math

Math

Reading and Writing

Skills

Example Questions

Example Question 1

Key: -42

Key Explanation

Example Question 2

Key: B

Key Explanation

Distractor Explanations