QCM : Algebraic Techniques for Equations — 11 questions

Explication

Algebraic substitution specifically refers to replacing a variable in an expression or equation with a known value or another expression, then simplifying to evaluate or manipulate the expression.

Explication

Substituting x = 4 into y = 2x + 3 gives y = 2(4) + 3 = 8 + 3 = 11. Therefore, the correct answer is 11.

Explication

The main purpose of a literal equation is to rearrange the formula to make any variable the subject, allowing for solving or substitution of that variable in different contexts.

Explication

The correct answer is 'After the formula was fully rearranged,' because this is the point at which the formula has been manipulated so that the variable of interest is isolated on one side, making it ready for substitution or evaluation. This step occurs after the rearrangement process is complete, which is essential for solving the problem.

Explication

Restrictions on variables specify the values that variables cannot take because they make an expression undefined or invalid, such as division by zero or square roots of negative numbers. In contrast, algebraic substitution is a process where variables are replaced with specific known values to evaluate or simplify expressions. These concepts serve different purposes: restrictions define the domain limitations, while substitution is a calculation technique.

Explication

René Descartes is credited with developing analytic geometry, which includes the study and formulation of linear functions as equations of straight lines.

Explication

Increasing the slope of a line makes it steeper, meaning it rises or falls more rapidly as it moves along the x-axis. This directly affects the line's steepness and the solutions to equations involving the line, such as the intersection points with other lines or axes.

Explication

The best approach is to solve one of the equations for one variable, such as x from the second equation, and then substitute that expression into the first. This is the substitution method, which is effective here because the second equation is already solved for x. The other options are plausible but less straightforward: graphing is valid but not an algebraic application; adding the equations would eliminate y, not x; and multiplying the second equation by 2 would not directly lead to an elimination of x or y without further steps.

Explication

The key feature of graphical methods is the graph of the line itself, which visually shows the relationship between variables, including slope, intercepts, and solutions.

Explication

The substitution method involves replacing a variable in one equation with an equivalent expression from another equation, then solving for the remaining variable. It is a systematic algebraic process used specifically for solving systems of equations.

Explication

The technique described involves adding or subtracting equations to eliminate a variable, which is specifically called the elimination method. This method is one of the standard approaches for solving systems of equations, as detailed in the content.