Question

$$ \begin{aligned} & 1.3 x-0.6 y=-0.7 \\ & 6.5 x-1.5 y=-0.5 \end{aligned} $$ 26\. When two equations above are graphed in the \(x y\)-plane, there is a single solution at \((x, y)\). What is the \(y\)-coordinate of that solution? A) \(-1.33\) B) \(-1.125\) C) 2 D) \(3.25\)

Short answer

\(y = 2\)

Step-by-step solution

Express \(x\) in terms of \(y\) using the first equation

From the first equation, we can solve for \(x\) in terms of \(y\) as follows: \(1.3x = 0.6y - 0.7\) \(x = \frac{0.6y - 0.7}{1.3}\)

Substitute the expression for \(x\) into the second equation

Now, substitute this expression for \(x\) into the second equation as follows: \(6.5 \cdot \frac{0.6y - 0.7}{1.3} - 1.5y = -0.5\)

Solve the equation for \(y\)

Now we will solve for \(y\): \(6.5(0.6y - 0.7) - 1.3(1.5y) = -0.65\) \(3.9y - 4.55 - 1.95y = -0.65\) \(1.95y = 3.9\) \(y = 2\)

Find the value of \(x\) using the \(y\)-coordinate

We found \(y=2\) in the previous step. Now, we can find the value of \(x\) by substituting this value into the expression we derived for \(x\) in terms of \(y\): \(x = \frac{0.6y - 0.7}{1.3}\) \(x = \frac{0.6(2) - 0.7}{1.3}\) \(x = \frac{1.2 - 0.7}{1.3}\) \(x = \frac{0.5}{1.3}\) So, when the two equations are graphed in the \(xy\)-plane, there is a single solution at \((x, y) = \left(\frac{0.5}{1.3}, 2\right)\). Thus, the \(y\)-coordinate of that solution is:

Answer

\(y = 2\), which corresponds to answer choice (C) in the multiple-choice question.

Key concepts

Systems of Equations

Understanding systems of equations is fundamental when tackling SAT math problems. A system consists of two or more equations with the same variables, and the solution is the point(s) where the equations intersect, representing the same values for the variables. Typically, you'll encounter linear systems, where each equation represents a straight line on a coordinate plane.

There are several methods to solve systems of equations: substitution, elimination, and graphical solution. The substitution method involves solving one equation for a variable and then substituting that expression into the other equation(s). Elimination involves adding or subtracting equations to eliminate one variable, making it easier to solve for the other. In the SAT context, recognizing which method is most efficient can save time and reduce calculation errors.

With the essential concept of systems of equations addressed, you'll often be required to interpret or construct these systems from word problems, making it a versatile tool in the SAT Math arsenal.

Graphical Solution Method

The graphical solution method is a visually intuitive approach to solving systems of equations. By graphing each equation on the same set of axes, the point where they intersect represents the solution to the system. This method is particularly useful for understanding the nature of the solutions: one solution, no solution, or infinitely many solutions.

When applying the graphical method, it's vital to accurately plot the lines based on the equation's slope and intercept. Remember that errors in drawing can lead to incorrect conclusions, so it's usually used as a supplement to algebraic methods on the SAT. Let's not forget that some SAT questions may provide graphs or require you to interpret them, so being comfortable with reading graphs is just as crucial as being able to plot them.

SAT Math Strategies

Achieving success in SAT Math is not just about knowing the content; it's equally about strategizing. Efficiently navigating the SAT Math section involves several key strategies. First, time management is crucial; knowing when to move on from a problem and to return to it later can make a significant difference.

Another strategy is to familiarize yourself with the structure and types of questions on the SAT. There are multiple-choice questions and grid-ins, and knowing how to approach each type can enhance your performance. For example, with multiple-choice questions, you can sometimes work backward from the answers or use process of elimination. Lastly, master the art of checking your work. Silly mistakes can cost you points, so if time permits, recheck your calculations and make sure your answers make sense within the context of the problem.

Implementing these SAT Math strategies, along with a thorough understanding of mathematical concepts, will increase your confidence and ability to score well on the test.